° | a | b | c | d |
a | {a} | {a} | {a} | {a} |
b | {a} | {a} | {a} | {a} |
c | {a} | {a} | {a} | {a, b} |
d | {a} | {a} | {a, b} | {a, b, c} |
Citation: Jan-Peter Hildebrandt. Pore-forming virulence factors of Staphylococcus aureus destabilize epithelial barriers-effects of alpha-toxin in the early phases of airway infection[J]. AIMS Microbiology, 2015, 1(1): 11-36. doi: 10.3934/microbiol.2015.1.11
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In the opinion of Zadeh [45], fuzzy set theory, which was introduced in 1965, says that decision-makers should take membership degree into account while settling ambiguous situations. It is a method of conveying and presenting vague or ill-defined information. The concept of fuzzy sets has been explored by several researchers (see, e.g., [9,11,38,42,46]). In mathematics, the concept of a fuzzy set is a generalization of classical sets. There are various extensions of fuzzy sets, such as intuitionistic fuzzy sets [4], Pythagorean fuzzy sets [44], Fermatean fuzzy sets [34], spherical fuzzy sets [3], picture fuzzy sets [7], and linear Diophantine fuzzy sets [33], among others. In this research, we will review the extensions of fuzzy sets relevant to this study, namely intuitionistic fuzzy sets, Pythagorean fuzzy sets, and Fermatean fuzzy sets. In 1986, Atanassov [4] introduced the notion of intuitionistic fuzzy sets as a generalization of fuzzy sets. These sets consist of an element's degree of membership and non-membership in a universe set, with the rule that sum of these degrees not be greater than one. Currently, the concept of intuitionistic fuzzy sets is still being studied continuously [10,22,23,41]. Subsequently, Yager [44] introduced the notion of Pythagorean fuzzy sets, where the sum of the squares of membership and non-membership is constrained to the unit interval $ [0, 1] $. This concept generalizes both fuzzy sets and intuitionistic fuzzy sets. In addition, Senapati and Yager [34] first introduced the concept of Fermatean fuzzy sets in 2019, defining them as the cube sum of their membership and non-membership degrees in $ [0, 1] $. The fuzzy sets, intuitionistic fuzzy sets, and Pythagorean fuzzy sets are all generalized by Fermatean fuzzy sets. For example, consider two real numbers, $ 0.7 $ and $ 0.8 $, in the interval $ [0, 1] $. We can observe that $ 0.7 + 0.8 > 1 $ and $ (0.7)^2 + (0.8)^2 > 1 $, but $ (0.7)^3 + (0.8)^3 < 1 $. This means that the Fermatean fuzzy sets have a better information space than the intuitionistic fuzzy sets and the Pythagorean fuzzy sets.
The concepts of various types of fuzzy set mentioned above are applied to the classes of algebras, helping develop the basic properties of these algebras. The semigroup is an essential structure in abstract algebra and has applications in automata theory, numerical theory, functional analysis, and optimization, among other mathematical and theoretical fields. The study of the regularity of semigroups is an important and trending area of research. This article will briefly review the classification of semigroups using various types of fuzzy sets. Kehayopulu and Tsingelis [18] used fuzzy quasi-ideals and fuzzy left (resp., right) ideals to characterize regular ordered semigroups. Xie and Tang [43] later developed fuzzy left (resp., right) ideals, fuzzy (resp., generalized) bi-ideals, and fuzzy quasi-ideals that characterized the classes of regular and intra-regular ordered semigroups. Further characterizations of regular, intra-regular, and left weakly regular ordered semigroups were then provided by Khan and Shabir [19], using their intuitionistic fuzzy left (resp., right) ideals. Subsequently, Hussain et al. [13] introduced the concept of rough Pythagorean fuzzy ideals in semigroups, which extends to the lower and upper approximations of bi-ideals, interior ideals, $ (1, 2) $-ideals, and Pythagorean fuzzy left (resp., right) ideals of semigroups. Afterwards, the concepts of Pythagorean fuzzy prime ideals and semi-prime ideals of ordered semigroups, together with some of the essential features of Pythagorean fuzzy regular and intra-regular ordered semigroup ideals, were examined by Adak et al. [2]. A review of relations is provided for the family of Fermatean fuzzy regular ideals of ordered semigroups, and Adak et al. [2] determined the concept of Fermatean fuzzy semi-prime (resp., prime) ideals. For using different types of fuzzy sets to classify the regularity of semigroups, see [5,17,20,21,36].
As a generalization to ordinary algebraic structures, Marty [24] gave algebraic hyperstructures in 1934. In an algebraic hyperstructure, the composition of two elements is a nonempty set, but in an ordinary algebraic structure, the composition of two elements is an element. The notion of a semigroup is generalized to form a semihypergroup. Several authors have investigated various facets of semihypergroups; for instance, see [8,12,31,32]. Fuzzy set theory gives a novel field of study called fuzzy hyperstructures. In 2014, Hila and Abdullah [16] characterized various classes of $ \Gamma $-semihypergroups using intuitionistic fuzzy left (resp., right, two-sided) $ \Gamma $-hyperideals and intuitionistic bi-$ \Gamma $-hyperideals. Afterwards, the characteristics of fuzzy quasi-$ \Gamma $-hyperideals were used by Tang et al. [39] in 2017 to study characterizations of regular and intra-regular ordered $ \Gamma $-semihypergroups. Additional characterizations of regular semihypergroups and intra-regular semihypergroups were given by Shabir et al. [35], based on the properties of their $ (\in, \in\vee q) $-bipolar fuzzy hyperideals and $ (\in, \in\vee q) $-bipolar fuzzy bi-hyperideals. Furthermore, Masmali [25] used Pythagorean picture fuzzy sets hyperideals to characterize the class of regular semihypergroups. More recently, Nakkhasen [28] introduced Fermatean fuzzy subsemihypergroups, Fermatean fuzzy (resp., left, right) hyperideals, and Fermatean fuzzy (resp., generalized) bi-hyperideals of semihypergroups in 2023. Additionally, some characterizations of regular semihypergroups were made using their corresponding types of Fermatean fuzzy hyperideals. Further, Nakkhasen has also studied the characterizations of different types of regularities in algebraic structures involving semigroups using the concept of generalized fuzzy sets, such as picture fuzzy sets, spherical fuzzy sets, and Pythagorean fuzzy sets, which can be found in the following references [26,27,29,30].
As previously mentioned, there are various types of regularities in algebra that are related to semigroups, such as regular, intra-regular, completely regular, left regular, right regular, and generalized regular. However, the most popular are the regular and intra-regular types. It is known that the algebraic structure of semihypergroups is an extension of semigroups and ordered semigroups. The objective of this research is to classify the regularity of semihypergroups using the properties of Fermatean fuzzy set theory. For usage in the following section, we review the fundamental ideas and features of Fermatean fuzzy sets in semihypergroups in Section 2. In Section 3, which is the main section of our paper, we characterize intra-regular semihypergroups by Fermatean fuzzy left (resp., right) hyperideals, and Fermatean (resp., generalized) bi-hyperideals. Additionally, the notion of Fermatean fuzzy interior hyperideals of semihypergroups is defined, and the class of intra-regular semihypergroups is characterized by Fermatean fuzzy interior hyperideals. Finally, Section 4 delves into the features of Fermatean fuzzy left (resp., right) hyperideals and Fermatean (resp., generalized) bi-hyperideals of semihypergroups, which are used to characterize both regular and intra-regular semihypergroups.
A map $ \circ:X\times X\rightarrow P^*(X) $ is called a hyperoperation (see [24]) on a nonempty set $ X $ where $ P^*(X) $ is the set of all nonempty subsets of $ X $. The pair $ (X, \circ) $ is called a hypergroupoid. Let $ X $ be a nonempty set and let $ A, B\in P^*(X) $ and $ x\in X $. Then, we denote
$ A\circ B = \bigcup\limits_{a\in A,b\in B}a\circ b, A\circ x = A\circ\{x\}\text{ and }x\circ B = \{x\}\circ B. $ |
A hypergroupoid $ (S, \circ) $ is said to be a semihypergroup (see [6]) if for every $ x, y, z\in S $, $ (x\circ y)\circ z = x\circ (y\circ z) $, which means that $ {\bigcup\limits_{u\in x\circ y}u\circ z = \bigcup\limits_{v\in y\circ z}x\circ v} $. For simplicity, we represent a semihypergroup as $ S $ instead of a semihypergroup $ (S, \circ) $, $ AB $ represents $ A\circ B $, for all nonempty subsets $ A $ and $ B $ of $ S $, and $ xy $ represents $ x\circ y $, for all $ x, y\in S $.
Now, we will review the notions of various types of hyperideals in semihypergroups, taken from [14] and [37]. A nonempty subset $ A $ of a semihypergroup $ S $ is called:
(ⅰ) a subsemihypergroup of $ S $ if $ AA\subseteq A $;
(ⅱ) a left hyperideal of $ S $ if $ SA\subseteq A $;
(ⅲ) a right hyperideal of $ S $ if $ AS\subseteq A $;
(ⅳ) a hyperideal of $ S $ if it is both a left and a right hyperideal of $ S $;
(ⅴ) a bi-hyperideal of $ S $ if $ AA\subseteq A $ and $ ASA\subseteq A $;
(ⅵ) a generalized bi-hyperideal of $ S $ if $ ASA\subseteq A $;
(ⅶ) an interior hyperideal of $ S $ if $ AA\subseteq A $ and $ SAS\subseteq A $.
A map $ f:X\rightarrow [0, 1] $ from a nonempty set $ X $ into the unit interval is called a fuzzy set [45]. Let $ f $ and $ g $ be any two fuzzy sets of a nonempty set $ X $. The notions $ f\cap g $ and $ f\cup g $ are defined by $ (f\cap g)(x) = \min\{f(x), g(x)\} $ and $ (f\cup g)(x) = \max\{f(x), g(x)\} $, for all $ x\in X $, respectively.
A Fermatean fuzzy set [34] (briefly, FFS) on a nonempty set $ X $ is defined as:
$ \mathcal{A}: = \{\langle x,\mu_{\mathcal{A}}(x),\lambda_{\mathcal{A}}(x)\rangle\mid x\in X\}, $ |
where $ \mu_{\mathcal{A}}:X\rightarrow [0, 1] $ and $ \lambda_{\mathcal{A}}:X\rightarrow [0, 1] $ represent the degree of membership and non-membership of each $ x\in X $ to the set $ \mathcal{A} $, respectively, with satisfy $ 0\leq (\mu_{\mathcal{A}}(x))^3+(\lambda_{\mathcal{A}}(x))^3\leq 1 $, for all $ x\in X $. Throughout this paper, we will use the symbol $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ instead of the FFS $ \mathcal{A} = \{\langle x, \mu_{\mathcal{A}}(x), \lambda_{\mathcal{A}}(x)\rangle\mid x\in X\} $.
In 2023, Nakkhasen [28] defined the concepts of many types of Fermatean fuzzy hyperideals in semihypergroups as follows. Let $ S $ be a semihypergroup, and $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ be an FFS on $ S $. Then:
(ⅰ) $ \mathcal{A} $ is called a Fermatean fuzzy subsemihypergroup (briefly, FFSub) of $ S $ if for every $ x, y\in S $,
$ {\inf\limits_{z\in xy}}\mu_{\mathcal{A}}(z)\geq \min\{\mu_{\mathcal{A}}(x), \mu_{\mathcal{A}}(y)\}\;{\rm{and}}\;{\sup\limits_{z\in xy}}\lambda_{\mathcal{A}}(z) \leq \max\{\lambda_{\mathcal{A}}(x), \lambda_{\mathcal{A}}(y)\};$ |
(ⅱ) $ \mathcal{A} $ is called a Fermatean fuzzy left hyperideal (briefly, FFL) of $ S $ if for every $ x, y\in S $,
$ {\inf\limits_{z\in xy}}\mu_{\mathcal{A}}(z)\geq \mu_{\mathcal{A}}(y)\;{\rm{and}}\;{\sup\limits_{z\in xy}}\lambda_{\mathcal{A}}(z)\leq \lambda_{\mathcal{A}}(y)$; |
(ⅲ) $ \mathcal{A} $ is called a Fermatean fuzzy right hyperideal (briefly, FFR) of $ S $ if for every $ x, y\in S $,
$ {\inf\limits_{z\in xy}}\mu_{\mathcal{A}}(z)\geq \mu_{\mathcal{A}}(x)\;{\rm{and}}\;{\sup\limits_{z\in xy}}\lambda_{\mathcal{A}}(z)\leq \lambda_{\mathcal{A}}(x)$; |
(ⅳ) $ \mathcal{A} $ is called a Fermatean fuzzy hyperideal (briefly, FFH) of $ S $ if it is both an FFL and an FFR of $ S $;
(ⅴ) an FFSub $ \mathcal{A} $ of $ S $ is called a Fermatean fuzzy bi-hyperideal (briefly, FFB) of $ S $ if for every $ w, x, y\in S $,
$ {\inf\limits_{z\in xwy}}\mu_{\mathcal{A}}(z)\geq \min\{\mu_{\mathcal{A}}(x), \mu_{\mathcal{A}}(y)\}\;{\rm{and}}\;{\sup\limits_{z\in xwy}}\lambda_{\mathcal{A}}(z)\leq \max\{\lambda_{\mathcal{A}}(x), \lambda_{\mathcal{A}}(y)\}$; |
(ⅵ) a FFS $ \mathcal{A} $ of $ S $ is called a Fermatean fuzzy generalized bi-hyperideal (briefly, FFGB) of $ S $ if for every $ w, x, y\in S $,
$ {\inf\limits_{z\in xwy}}\mu_{\mathcal{A}}(z)\geq \min\{\mu_{\mathcal{A}}(x), \mu_{\mathcal{A}}(y)\}\;{\rm{and}}\;{\sup\limits_{z\in xwy}}\lambda_{\mathcal{A}}(z)\leq \max\{\lambda_{\mathcal{A}}(x), \lambda_{\mathcal{A}}(y)\}$. |
For any FFSs $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ and $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ on a nonempty set $ X $, we denote:
(ⅰ) $ \mathcal{A}\subseteq \mathcal{B} $ if and only if $ \mu_{\mathcal{A}}(x)\leq\mu_{\mathcal{B}}(x) $ and $ \lambda_{\mathcal{A}}(x)\geq \lambda_{\mathcal{B}}(x) $, for all $ x\in X $;
(ⅱ) $ \mathcal{A} = \mathcal{B} $ if and only if $ \mathcal{A}\subseteq \mathcal{B} $ and $ \mathcal{B}\subseteq \mathcal{A} $;
(ⅲ) $ \mathcal{A}\cap\mathcal{B}: = \{\langle x, (\mu_{\mathcal{A}}\cap\mu_{\mathcal{B}})(x), (\lambda_{\mathcal{A}}\cup\lambda_{\mathcal{B}})(x)\rangle\mid x\in X \} $;
(ⅳ) $ \mathcal{A}\cup\mathcal{B}: = \{\langle x, (\mu_{\mathcal{A}}\cup\mu_{\mathcal{B}})(x), (\lambda_{\mathcal{A}}\cap\lambda_{\mathcal{B}})(x)\rangle\mid x\in X \} $.
We observe that $ \mathcal{A}\cap\mathcal{B} $ and $ \mathcal{A}\cup\mathcal{B} $ are FFSs of $ X $ if $ \mathcal{A} $ and $ \mathcal{B} $ are FFSs on $ X $.
Let $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ and $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ be any FFSs of a semihypergroup $ S $. Then, the Fermatean fuzzy product of $ \mathcal{A} $ and $ \mathcal{B} $ is defined as
$ \mathcal{A}\circ\mathcal{B}: = \{\langle x, (\mu_{\mathcal{A}}\circ\mu_{\mathcal{B}})(x), (\lambda_{\mathcal{A}}\circ\lambda_{\mathcal{B}})(x)\rangle \mid x\in S\}, $ |
where
$ (\mu_{\mathcal{A}}\circ\mu_{\mathcal{B}})(x) = {supx∈ab[min{μA(a),μB(b)}]if x∈S2,0otherwise, $
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$ (\lambda_{\mathcal{A}}\circ\lambda_{\mathcal{B}})(x) = {infx∈ab[max{λA(a),λB(b)}]if x∈S2,1otherwise. $
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For any semihypergroup $ S $, we determine the FFSs $ \mathcal{S}: = \{\langle x, 1, 0\rangle\mid x\in S\} $ and $ \mathcal{O}: = \{\langle x, 0, 1\rangle\mid x\in S\} $ on $ S $. This obtains that $ \mathcal{A}\subseteq \mathcal{S} $ and $ \mathcal{O}\subseteq \mathcal{A} $, for all FFS $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ on $ S $. The Fermatean characteristic function of a subset $ A $ of a semihypergroup $ S $, as an FFS on $ S $, defined by $ \mathcal{C}_A = \{\langle x, \mu_{\mathcal{C}_A}(x), \lambda_{\mathcal{C}_A}(x)\rangle\mid x\in S\} $, where
$\mu_{\mathcal{C}_{A}}(x) = {1if x∈A,0otherwise, \;{\rm{and}}\;\lambda_{\mathcal{C}_{A}}(x) = {0if x∈A,1otherwise. $
|
We note that if for each subset $ A $ of $ S $ such that $ A = S $ (resp., $ A = \emptyset $), then $ \mathcal{C}_A = \mathcal{S} $ (resp., $ \mathcal{C}_A = \mathcal{O} $). All the above-mentioned notions are presented in [28].
Lemma 2.1. [28] Let $ \mathcal{C}_A = (\mu_{\mathcal{C}_A}, \lambda_{\mathcal{C}_A}) $ and $ \mathcal{C}_B = (\mu_{\mathcal{C}_{B}}, \lambda_{\mathcal{C}_B}) $ be FFSs of a semihypergroup $ S $ with respect to nonempty subsets $ A $ and $ B $ of $ S $, respectively. Then the following axioms hold:
(ⅰ) $ \mathcal{C}_{A\cap B} = \mathcal{C}_A\cap\mathcal{C}_B $;
(ⅱ) $ \mathcal{C}_{AB} = \mathcal{C}_A\circ\mathcal{C}_B $.
Lemma 2.2. [28] Let $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $, $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $, $ \mathcal{C} = (\mu_{\mathcal{C}}, \lambda_{\mathcal{C}}) $ and $ \mathcal{D} = (\mu_{\mathcal{D}}, \lambda_{\mathcal{D}}) $ be any FFSs of a semihypergroup $ S $. If $ \mathcal{A}\subseteq \mathcal{B} $ and $ \mathcal{C}\subseteq \mathcal{D} $, then $ \mathcal{A}\circ\mathcal{C}\subseteq \mathcal{B}\circ\mathcal{D} $.
Lemma 2.3. [28] Let $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ be an FFS on a semihypergroup $ S $. The following conditions hold:
(ⅰ) $ \mathcal{A} $ is an FFSub of $ S $ if and only if $ \mathcal{A}\circ\mathcal{A}\subseteq \mathcal{A} $;
(ⅱ) $ \mathcal{A} $ is an FFL of $ S $ if and only if $ \mathcal{S}\circ\mathcal{A}\subseteq \mathcal{A} $;
(ⅲ) $ \mathcal{A} $ is an FFR of $ S $ if and only if $ \mathcal{A}\circ\mathcal{S}\subseteq \mathcal{A} $;
(ⅳ) $ \mathcal{A} $ is an FFGB of $ S $ if and only if $ \mathcal{A}\circ\mathcal{S}\circ\mathcal{A}\subseteq \mathcal{A} $;
(ⅴ) $ \mathcal{A} $ is an FFB of $ S $ if and only if $ \mathcal{A}\circ\mathcal{A}\subseteq \mathcal{A} $ and $ \mathcal{A}\circ\mathcal{S}\circ\mathcal{A}\subseteq \mathcal{A} $.
Lemma 2.4. [28] For any nonempty subset $ A $ of a semihypergroup $ S $, the following statements hold:
(i) $ A $ is a subsemihypergroup of $ S $ if and only if $ \mathcal{C}_A = (\mu_{\mathcal{C}_A}, \lambda_{\mathcal{C}_A}) $ is an FFSub of $ S $;
(ii) $ A $ is a left hyperideal of $ S $ if and only if $ \mathcal{C}_A = (\mu_{\mathcal{C}_A}, \lambda_{\mathcal{C}_A}) $ is an FFL of $ S $;
(iii) $ A $ is a right hyperideal of $ S $ if and only if $ \mathcal{C}_A = (\mu_{\mathcal{C}_A}, \lambda_{\mathcal{C}_A}) $ is an FFR of $ S $;
(iv) $ A $ is a hyperideal of $ S $ if and only if $ \mathcal{C}_A = (\mu_{\mathcal{C}_A}, \lambda_{\mathcal{C}_A}) $ is an FFH of $ S $;
(v) $ A $ is a generalized bi-hyperideal of $ S $ if and only if $ \mathcal{C}_A = (\mu_{\mathcal{C}_A}, \lambda_{\mathcal{C}_A}) $ is an FFGB of $ S $;
(vi) $ A $ is a bi-hyperideal of $ S $ if and only if $ \mathcal{C}_A = (\mu_{\mathcal{C}_A}, \lambda_{\mathcal{C}_A}) $ is an FFB of $ S $.
A semihypergroup $ S $ is called regular (see [15]) if for every element $ a $ in $ S $, there exists $ x\in S $ such that $ a\in axa $. Equivalently, $ a\in aSa $, for all $ a\in S $ or $ A\subseteq ASA $, for any $ A\subseteq S $. A semihypergroup $ S $ is called intra-regular (see [35]) if, for any element $ a $ in $ S $, there exist $ x, y\in S $ such that $ a\in xa^2y $. Equivalently, $ a\in Sa^2S $, for all $ a\in S $ or $ A\subseteq SA^2S $, for each $ A\subseteq S $.
Example 2.5. Let $ \mathbb{N} $ denote the set of all natural numbers. Define a hyperoperation $ \circ $ on $ \mathbb{N} $ by $ a\circ b: = \{x\in \mathbb{N}\mid x\leq ab\} $, for all $ a, b\in\mathbb{N} $. Next, we claim that the hyperoperation $ \circ $ on $ \mathbb{N} $ is consistent with the associative property. Let $ a, b\in\mathbb{N} $ and $ x\in (a\circ b)\circ c $. Then, $ x\in u\circ c $, for some $ u\in a\circ b $. So, $ x\leq uc $ and $ u\leq ab $. It follows that $ x\leq uc\leq (ab)c = a(bc) $. Also, $ x\in a\circ(bc)\subseteq a\circ(b\circ c) $, since $ bc\in b\circ c $. Thus, $ (a\circ b)\circ c\subseteq a\circ (b\circ c) $. Similarly, we can prove that $ a\circ (b\circ c)\subseteq (a\circ b)\circ c $. Hence, $ (a\circ b)\circ c = a\circ (b\circ c) $. Therefore, $ (\mathbb{N}, \circ) $ is a semihypergroup. Now, for every $ a\in \mathbb{N} $, we have $ a\leq axa $ and $ a\leq ya^2z $, for some $ x, y, z\in \mathbb{N} $. This implies that $ a\in a\circ x\circ a $ and $ a\in y\circ a\circ a\circ z $. It turns out that $ (\mathbb{N}, \circ) $ is a regular and intra-reular semihypergroup.
Lemma 2.6. [28] Let $ S $ be a semihypergroup. Then, $ S $ is regular if and only if $ \mathcal{R}\cap\mathcal{L} = \mathcal{R}\circ\mathcal{L} $, for any FFR $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ and any FFL $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ of $ S $.
Lemma 2.7. [35] Let $ S $ be a semihypergroup. Then, $ S $ is intra-regular if and only if $ L\cap R\subseteq LR $, for every left hyperideal $ L $ and every right hyperideal $ R $ of $ S $.
In this section, we present results about the characterizations of intra-regular semihypergroups by properties of FFLs, FFRs, FFBs, and FFGBs of semihypergroups.
Theorem 3.1. Let $ S $ be a semihypergroup. Then, $ S $ is intra-regular if and only if $ \mathcal{L}\cap\mathcal{R}\subseteq \mathcal{L}\circ\mathcal{R} $, for every FFL $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ and every FFR $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ of $ S $.
Proof. Assume that $ S $ is intra-regular. Let $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ and $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ be an FFL and an FFR of $ S $, respectively. For any $ a\in S $, there exist $ x, y\in S $ such that $ a\in xa^2y $. Then, we have
$ (μL∘μR)(a)=supa∈pq[min{μL(p),μR(q)}]≥min{infp∈xaμL(p),infq∈ayμR(q)}≥min{μL(a),μR(a)}=(μL∩μR)(a), $
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and
$ (λL∘λR)(a)=infa∈pq[max{λL(p),λR(q)}]≤max{supp∈xaλL(p),supq∈ayλR(q)}≤max{λL(a),λR(a)}=(λL∪λR)(a). $
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Hence, $ \mathcal{L}\cap\mathcal{R}\subseteq \mathcal{L}\circ\mathcal{R} $.
Conversely, let $ L $ and $ R $ be any left hyperideal and any right hyperideal of $ S $, respectively. By Lemma 2.4, we have $ \mathcal{C}_L = (\mu_{\mathcal{C}_L}, \lambda_{\mathcal{C}_L}) $ and $ \mathcal{C}_R = (\mu_{\mathcal{C}_R}, \lambda_{\mathcal{C}_R}) $ are an FFL and an FFR of $ S $, respectively. By the given assumption and Lemma 2.1, we get
$ \mathcal{C}_{L\cap R} = \mathcal{C}_L\cap\mathcal{C}_R\subseteq \mathcal{C}_L\circ\mathcal{C}_R = \mathcal{C}_{LR}. $ |
Now, let $ a\in L\cap R $. Thus, we have $ \mu_{\mathcal{C}_{LR}}(a)\geq \mu_{\mathcal{C}_{L\cap R}}(a) = 1 $. Also, $ \mu_{\mathcal{C}_{LR}}(a) = 1 $; that is, $ a\in LR $. This implies that $ L\cap R\subseteq LR $. By Lemma 2.7, we conclude that $ S $ is intra-regular.
Theorem 3.2. Let $ S $ be a semihypergroup. Then the following statements are equivalent:
(ⅰ) $ S $ is intra-regular;
(ⅱ) $ \mathcal{L}\cap\mathcal{G}\subseteq \mathcal{L}\circ\mathcal{G}\circ\mathcal{S} $, for each FFL $ \mathcal{L} = (\mu_\mathcal{L}, \lambda_{\mathcal{L}}) $ and each FFGB $ \mathcal{G} = (\mu_\mathcal{G}, \lambda_{\mathcal{G}}) $ of $ S $;
(ⅲ) $ \mathcal{L}\cap\mathcal{B}\subseteq \mathcal{L}\circ\mathcal{B}\circ\mathcal{S} $, for each FFL $ \mathcal{L} = (\mu_\mathcal{L}, \lambda_{\mathcal{L}}) $ and each FFB $ \mathcal{B} = (\mu_\mathcal{B}, \lambda_{\mathcal{B}}) $ of $ S $.
Proof. (ⅰ)$ \Rightarrow $(ⅱ) Assume that $ S $ is intra-regular. Let $ \mathcal{L} = (\mu_\mathcal{L}, \lambda_{\mathcal{L}}) $ and $ \mathcal{G} = (\mu_\mathcal{G}, \lambda_{\mathcal{G}}) $ be an FFL and an FFGB of $ S $, respectively. Let $ a\in S $. Then, there exist $ x, y\in S $ such that $ a\in xa^2y $. It follows that $ a\in (x^2a)(ayay) $. Thus, we have
$ (μL∘μG∘μS)(a)=supa∈pq[min{μL(p),(μG∘μS)(q)}]=supa∈pq[min{μL(p),supq∈mn[min{μG(m),μS(n)}]}]≥min{infp∈x2aμL(p),min{infm∈ayaμG(m),μS(y)}}≥min{μL(a),min{μG(a),μG(a)}}=min{μL(a),μG(a)}=(μL∩μG)(a), $
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and
$ (λL∘λG∘λS)(a)=infa∈pq[max{λL(p),(λG∘λS)(q)}]=infa∈pq[max{λL(p),infq∈mn[max{λG(m),λS(n)}]}]≤max{supp∈x2aλL(p),max{supm∈ayaλG(m),λS(y)}}≤max{λL(a),max{λG(a),λG(a)}}=max{λL(a),λG(a)}=(λL∪λG)(a). $
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This means that $ \mathcal{L}\cap\mathcal{G}\subseteq \mathcal{L}\circ\mathcal{G}\circ\mathcal{S} $.
(ⅱ)$ \Rightarrow $(ⅲ) Since every FFB is also an FFGB of $ S $, it follows that (ⅲ) holds.
(ⅲ)$ \Rightarrow $(ⅰ) Let $ \mathcal{L} = (\mu_\mathcal{L}, \lambda_{\mathcal{L}}) $ and $ \mathcal{R} = (\mu_\mathcal{R}, \lambda_{\mathcal{R}}) $ be an FFL and an FFR of $ S $, respectively. We obtain that $ R $ is also an FFB of $ S $. By assumption, we have $ \mathcal{L}\cap\mathcal{R}\subseteq \mathcal{L}\circ(\mathcal{R}\circ\mathcal{S})\subseteq \mathcal{L}\circ\mathcal{R} $. By Theorem 3.1, it turns out that $ S $ is intra-regular.
Theorem 3.3. Let $ S $ be a semihypergroup. Then the following statements are equivalent:
(ⅰ) $ S $ is intra-regular;
(ⅱ) $ \mathcal{G}\cap\mathcal{R}\subseteq \mathcal{S}\circ\mathcal{G}\circ\mathcal{R} $, for each FFR $ \mathcal{R} = (\mu_\mathcal{R}, \lambda_{\mathcal{R}}) $ and each FFGB $ \mathcal{G} = (\mu_\mathcal{G}, \lambda_{\mathcal{G}}) $ of $ S $;
(ⅲ) $ \mathcal{B}\cap\mathcal{R}\subseteq \mathcal{S}\circ\mathcal{B}\circ\mathcal{R} $, for each FFR $ \mathcal{R} = (\mu_\mathcal{R}, \lambda_{\mathcal{R}}) $ and each FFB $ \mathcal{B} = (\mu_\mathcal{B}, \lambda_{\mathcal{B}}) $ of $ S $.
Proof. (ⅰ)$ \Rightarrow $(ⅱ) Assume that $ S $ is intra-regular. Let $ a\in S $. Then, there exist $ x, y\in S $ such that $ a\in (xaxa)(ay^2) $. Hence, we have
$ (μS∘μG∘μR)(a)=supa∈pq[min{(μS∘μG)(p),μR(q)}]=supa∈pq[min{supp∈mn[min{μS(m),μG(n)}],μR(q)}]≥min{min{μS(x),infn∈axaμG(n)},infq∈ay2μR(q)}≥min{min{μG(a),μG(a)},μR(a)}=min{μG(a),μR(a)}=(μR∩μG)(a), $
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and
$ (λS∘λG∘λR)(a)=infa∈pq[max{(λS∘λG)(p),λR(q)}]=infa∈pq[max{infp∈mn[max{λS(m),λG(n)}],λR(q)}]≤max{max{λS(x),supn∈axaλG(n)},supq∈ay2λR(q)}≤max{max{λG(a),λG(a)},λR(a)}=max{λG(a),λR(a)}=(λR∪λG)(a). $
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This shows that $ \mathcal{R}\cap\mathcal{G}\subseteq \mathcal{S}\circ\mathcal{G}\circ\mathcal{R} $.
(ⅱ)$ \Rightarrow $(ⅲ) Since every FFB is also an FFGB of $ S $, it is well done.
(ⅲ)$ \Rightarrow $(ⅰ) Let $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ and $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ be an FFL and an FFR of $ S $, respectively. Then, $ \mathcal{L} $ is also an FFB of $ S $. By the hypothesis, we have $ \mathcal{L}\cap\mathcal{R}\subseteq (\mathcal{L}\circ\mathcal{S})\circ\mathcal{R} \subseteq \mathcal{L}\circ\mathcal{R} $. By Theorem 3.1, we obtain that $ S $ is intra-regular.
Theorem 3.4. The following statements are equivalent in a semihypergroup $ S $:
(ⅰ) $ S $ is intra-regular;
(ⅱ) $ \mathcal{G}_1\cap\mathcal{G}_2\subseteq \mathcal{S}\circ\mathcal{G}_1\circ\mathcal{G}_2\circ\mathcal{S} $, for any FFGBs $ \mathcal{G}_1 = (\mu_{\mathcal{G}_1}, \lambda_{\mathcal{G}_1}) $ and $ \mathcal{G}_2 = (\mu_{\mathcal{G}_2}, \lambda_{\mathcal{G}_2}) $ of $ S $;
(ⅲ) $ \mathcal{B}_1\cap\mathcal{B}_2\subseteq \mathcal{S}\circ\mathcal{B}_1\circ\mathcal{B}_2\circ\mathcal{S} $, for any FFBs $ \mathcal{B}_1 = (\mu_{\mathcal{B}_1}, \lambda_{\mathcal{B}_1}) $ and $ \mathcal{B}_2 = (\mu_{\mathcal{B}_2}, \lambda_{\mathcal{B}_2}) $ of $ S $.
Proof. (ⅰ)$ \Rightarrow $(ⅱ) Let $ a\in S $. Then, there exist $ x, y\in S $ such that $ a\in xa^2y $. Thus, we have
$ (μS∘μG1∘μG2∘μS)(a)=supa∈pq[min{(μS∘μG1)(p),(μG2∘μS)(q)}]=supa∈pq[min{supp∈mn[min{μS(m),μG1(n)}],supq∈kl[min{μG2(k),μS(l)}]}]≥min{min{μS(x),μG1(a)},min{μG2(a),μS(y)}}=min{μG1(a),μG2(a)}=(μG1∩μG2)(a), $
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and
$ (λS∘λG1∘λG2∘λS)(a)=infa∈pq[max{(λS∘λG1)(p),(λG2∘λS)(q)}]=infa∈pq[max{infp∈mn[max{λS(m),λG1(n)}],infq∈kl[max{λG2(k),λS(l)}]}]≤max{max{λS(x),λG1(a)},max{λG2(a),λS(y)}}=max{λG1(a),λG2(a)}=(λG1∪λG2)(a). $
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This implies that $ \mathcal{G}_1\cap\mathcal{G}_2\subseteq \mathcal{S}\circ\mathcal{G}_1\circ\mathcal{G}_2\circ\mathcal{S} $.
(ⅱ)$ \Rightarrow $(ⅲ) It is obvious.
(ⅲ)$ \Rightarrow $(ⅰ) Let $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ be any FFL of $ S $, and $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ be any FFR of $ S $. Then, $ \mathcal{L} $ and $ \mathcal{R} $ are also FFBs of $ S $. By the hypothesis, we have $ \mathcal{L}\cap\mathcal{R}\subseteq (\mathcal{S}\circ\mathcal{L})\circ(\mathcal{R}\circ\mathcal{S})\subseteq \mathcal{L}\circ\mathcal{R} $. By Theorem 3.1, it follows that $ S $ is intra-regular.
Corollary 3.5. Let $ S $ be a semihypergroup. Then, the following conditions are equivalent:
(ⅰ) $ S $ is intra-regular;
(ⅱ) $ \mathcal{G}\subseteq \mathcal{S}\circ\mathcal{G}\circ\mathcal{G}\circ\mathcal{S} $, for any FFGB $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ of $ S $;
(ⅲ) $ \mathcal{B}\subseteq \mathcal{S}\circ\mathcal{B}\circ\mathcal{B}\circ\mathcal{S} $, for any FFB $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ of $ S $.
Proof. (ⅰ)$ \Rightarrow $(ⅱ) It follows by Theorem 3.4.
(ⅱ)$ \Rightarrow $(ⅲ) It is clear.
(ⅲ)$ \Rightarrow $(ⅰ) Let $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ and $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ be an FFL and an FFR of $ S $, respectively. It is not difficult to see that $ \mathcal{L}\cap\mathcal{R} $ is also an FFB of $ S $. By the given assumption, we have $ \mathcal{L}\cap\mathcal{R}\subseteq \mathcal{S}\circ(\mathcal{L}\cap\mathcal{R})\circ(\mathcal{L}\cap\mathcal{R}) \circ\mathcal{S}\subseteq (\mathcal{S}\circ\mathcal{L})\circ(\mathcal{R}\circ\mathcal{S})\subseteq \mathcal{L}\circ\mathcal{R} $. By Theorem 3.1, we conclude $ S $ is intra-regular.
The following corollary is obtained by Corollary 3.5.
Corollary 3.6. Let $ S $ be a semihypergroup. Then, $ S $ is intra-regular if and only if $ \mathcal{B}\cap\mathcal{G}\subseteq (\mathcal{S}\circ\mathcal{B}\circ\mathcal{B}\circ\mathcal{S}) \cap(\mathcal{S}\circ\mathcal{G}\circ\mathcal{G}\circ\mathcal{S}) $, for every FFB $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ and every FFGB $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ of $ S $.
Theorem 3.7. If $ S $ is an intra-regular semihypergroup, then $ \mathcal{A}\cap\mathcal{B} = \mathcal{A}\circ\mathcal{B} $, for each FFHs $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ and $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ of $ S $.
Proof. Assume that $ S $ is an intra-regular semihypergroup. Let $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ and $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ be FFHs of $ S $. Then, $ \mathcal{A}\circ\mathcal{B}\subseteq \mathcal{A}\circ\mathcal{S}\subseteq \mathcal{A} $ and $ \mathcal{A}\circ\mathcal{B}\subseteq \mathcal{S}\circ\mathcal{B}\subseteq \mathcal{B} $, it follows that $ \mathcal{A}\circ\mathcal{B}\subseteq \mathcal{A}\cap\mathcal{B} $. Next, let $ a\in S $. By assumption, there exist $ x, y\in S $ such that $ a\in xa^2y = (xa)(ay) $; that is, $ a\in pq $, for some $ p\in xa $ and $ q\in ay $. Thus, we have
$ (μA∘μB)(a)=supa∈pq[min{μA(p),μB(q)}]≥min{infp∈xaμA(p),infq∈ayμB(q)}≥min{μA(a),μB(a)}=(μA∩μB)(a), $
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and
$ (λA∘λB)(a)=infa∈pq[max{λA(p),λB(q)}]≤max{supp∈xaλA(p),supq∈ayλB(q)}≤max{λA(a),λB(a)}=(λA∪λB)(a). $
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Hence, $ \mathcal{A}\cap\mathcal{B}\subseteq \mathcal{A}\circ\mathcal{B} $. Therefore, $ \mathcal{A}\cap\mathcal{B} = \mathcal{A}\circ\mathcal{B} $.
Theorem 3.8. Let $ S $ be a semihypergroup. Then the following properties are equivalent:
(ⅰ) $ S $ is intra-regular;
(ⅱ) $ \mathcal{L}\cap\mathcal{G}\cap\mathcal{R}\subseteq \mathcal{L}\circ\mathcal{G}\circ\mathcal{R} $, for every FFL $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $, every FFR $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ and every FFGB $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ of $ S $;
(ⅲ) $ \mathcal{L}\cap\mathcal{B}\cap\mathcal{R}\subseteq \mathcal{L}\circ\mathcal{B}\circ\mathcal{R} $, for every FFL $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $, every FFR $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ and every FFB $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ of $ S $.
Proof. (ⅰ)$ \Rightarrow $(ⅱ) Assume that $ S $ is intra-regular. Let $ a\in S $. Then, there exist $ x, y\in S $ such that $ a\in xa^2y $, which implies that $ a\in (x^2a)(ayxaxa)(ay^3) $. Thus, $ a\in uvq $, for some $ u\in x^2a $, $ v\in ayxaxa $ and $ q\in ay^3 $. Also, there exists $ p\in S $ such that $ p\in uv $, and so $ a\in pq $. So, we have
$ (μL∘μG∘μR)(a)=supa∈pq[min{(μL∘μG)(p),μR(q)}]=supa∈pq[min{supp∈uv[min{μL(u),μG(v)}],μR(q)}]≥min{min{infu∈x2aμL(u),infv∈ayxaxaμG(v)},infq∈ay3μR(q)}≥min{min{μL(a),min{μG(a),μG(a)}},μR(a)}=min{μL(a),μG(a),μR(a)}=(μL∩μG∩μR)(a), $
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and
$ (λL∘λG∘λR)(a)=infa∈pq[max{(λL∘λG)(p),λR(q)}]=infa∈pq[max{infp∈uv[max{λL(u),λG(v)}],λR(q)}]≤max{max{supu∈x2aλL(u),supv∈ayxaxaλG(v)},supq∈ay3λR(q)}≤max{max{λL(a),max{λG(a),λG(a)}},λR(a)}=max{λL(a),λG(a),λR(a)}=(λL∪λG∪λR)(a). $
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This shows that $ \mathcal{L}\cap\mathcal{G}\cap\mathcal{R}\subseteq \mathcal{L}\circ\mathcal{G}\circ\mathcal{R} $.
(ⅱ)$ \Rightarrow $(ⅲ) It is clear.
(ⅲ)$ \Rightarrow $(ⅰ) Let $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ and $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ be any FFL and any FFR of $ S $, respectively. Then, $ \mathcal{R} $ is also an FFB of $ S $. By the given assumption, we have $ \mathcal{L}\cap\mathcal{R} = \mathcal{L}\cap\mathcal{R}\cap\mathcal{R}\subseteq \mathcal{L}\circ\mathcal{R}\circ\mathcal{R} \subseteq \mathcal{L}\circ\mathcal{R} $. By Theorem 3.1, we get that $ S $ is intra-regular.
Now, we introduce the notion of Fermatean fuzzy interior hyperideals in semihypergroups and investigate some properties of this notion. Moreover, we use the properties of Fermatean fuzzy interior hyperideals to study the characterizations of intra-regular semihypergroups.
Definition 3.9. An FFsub $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ is said to be a Fermatean fuzzy interior hyperideal (briefly, FFInt) of a semihypergroup $ S $ if for every $ w, x, y\in S $, $ {\inf\limits_{z\in wxy}\mu_{\mathcal{A}}(z)\geq \mu_{\mathcal{A}}(x)} $ and $ {\sup\limits_{z\in wxy}\lambda_{\mathcal{A}}(z)\leq\lambda_{\mathcal{A}}(x)} $.
Theorem 3.10. Let $ S $ be a semihypergroup, and $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ be an FFS of $ S $. Then, $ \mathcal{A} $ is an FFInt of $ S $ if and only if $ \mathcal{A}\circ\mathcal{A}\subseteq \mathcal{A} $ and $ \mathcal{S}\circ\mathcal{A}\circ\mathcal{S}\subseteq \mathcal{A} $.
Proof. Assume that $ \mathcal{A} $ is an FFInt of $ S $. Then, $ \mathcal{A} $ is an FFSub of $ S $. By Lemma 2.3, we have $ \mathcal{A}\circ\mathcal{A}\subseteq \mathcal{A} $. Now, let $ a\in S $. If $ a\not\in bcd $, for all $ b, c, d\in S $, then $ \mathcal{S}\circ\mathcal{A}\circ\mathcal{S}\subseteq \mathcal{A} $. Suppose that there exist $ p, q, x, y\in S $ such that $ a\in xy $ and $ x\in pq $. It follows that $ a\in pqy $. Thus, we have
$ (μS∘μA∘μS)(a)=supa∈xy[min{(μS∘μA)(x),μS(y)}]=supa∈xy[(μS∘μA)(x)]=supa∈xy[supx∈pq[min{μS(p),μA(q)}]]=supa∈xy[supx∈pq[μA(q)]]≤μA(a), $
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and
$ (λS∘λA∘λS)(a)=infa∈xy[max{(λS∘λA)(x),λS(y)}]=infa∈xy[(λS∘λA)(x)]=infa∈xy[infx∈pq[max{λS(p),λA(q)}]]=infa∈xy[infx∈pq[λA(q)]]≥λA(a). $
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Hence, $ \mathcal{S}\circ\mathcal{A}\circ\mathcal{S}\subseteq \mathcal{A} $. Conversely, let $ x, y, z\in S $, and let $ w\in xyz $. Then, there exists $ u\in xy $ such that $ w\in uz $. By assumption, we have
$ μA(w)≥(μS∘μA∘μS)=supw∈pq[min{(μS∘μA)(p),μS(q)}]≥{(μS∘μA)(u),μS(z)}=supu∈st[min{μS(s),μA(t)}]≥min{μS(x),μA(y)}=μA(y), $
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and
$ λA(w)≤(λS∘λA∘λS)=infw∈pq[max{(λS∘λA)(p),λS(q)}]≤{(λS∘λA)(u),λS(z)}=infu∈st[max{λS(s),λA(t)}]≤max{λS(x),λA(y)}=λA(y). $
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This shows that $ \mu_{\mathcal{A}}(w)\geq\mu_{\mathcal{A}}(y) $ and $ \lambda_{\mathcal{A}}(w)\leq\lambda_{\mathcal{A}}(y) $, for all $ w\in xyz $. It follows that $ {\inf\limits_{w\in xyz}}\mu_{\mathcal{A}}(z)\geq \mu_{\mathcal{A}}(y) $ and $ {\sup\limits_{w\in xyz}}\lambda_{\mathcal{A}}(z)\leq \lambda_{\mathcal{A}}(y) $. Therefore, $ \mathcal{A} $ is an FFInt of $ S $.
Theorem 3.11. Let $ S $ be a semihypergroup, and $ A $ be a nonempty subset of $ S $. Then, $ A $ is an interior hyperideal of $ S $ if and only if $ \mathcal{C}_A = (\mu_{\mathcal{C}_A}, \lambda_{\mathcal{C}_A}) $ is an FFInt of $ S $.
Proof. Assume that $ A $ is an interior hyperideal of $ S $. Then $ A $ is a subsemihypergroup of $ S $. By Lemma 2.4, we have $ \mathcal{C}_A $ is an FFSub of $ S $. Now, let $ x, y, z\in S $. If $ y\not\in A $, then $ \inf_{w\in xyz}\mu_{\mathcal{C}_A}(w)\geq 0 = \mu_{\mathcal{C}_A}(y) $ and $ \sup_{w\in xyz}\lambda_{\mathcal{C}_A}(w)\leq 1 = \lambda_{\mathcal{C}_A}(y) $. On the other hand, suppose that $ y\in A $. Thus, $ xyz\subseteq A $, which implies that for every $ w\in xyz $, we have $ \mu_{\mathcal{C}_A}(w) = 1 $ and $ \lambda_{\mathcal{C}_A}(w) = 0 $. This means that $ \mu_{\mathcal{C}_A}(w)\geq \mu_{\mathcal{C}_A}(y) $ and $ \lambda_{\mathcal{C}_A}(w)\leq\lambda_{\mathcal{C}_A}(y) $, for all $ w\in xyz $. That is, $ \inf_{w\in xyz}\mu_{\mathcal{C}_A}(w)\geq \mu_{\mathcal{C}_A}(y) $ and $ \sup_{w\in xyz}\lambda_{\mathcal{C}_A}(w)\leq \lambda_{\mathcal{C}_A}(y) $. Hence, $ \mathcal{C}_A $ is an FFInt of $ S $.
Conversely, assume that $ \mathcal{C}_A = (\mu_{\mathcal{C}_A}, \lambda_{\mathcal{C}_A}) $ is an FFInt of $ S $. Then, $ \mathcal{C}_A $ is an FFSub of $ S $. By Lemma 2.4, we have that $ A $ is a subsemihypergroup of $ S $. Let $ x, z\in S $ and $ y\in A $. By assumption, we get $ \inf_{w\in xyz}\mu_{\mathcal{C}_A}(w)\geq \mu_{\mathcal{C}_A}(y) = 1 $ and $ \sup_{w\in xyz}\lambda_{\mathcal{C}_A}(w)\leq \lambda_{\mathcal{C}_A}(y) = 0 $. This implies that $ \mu_{\mathcal{C}_A}(w)\geq 1 $ and $ \lambda_{\mathcal{C}_A}(w)\leq 0 $, for all $ w\in xyz $. Otherwise, $ \mu_{\mathcal{C}_A}(w)\leq 1 $ and $ \lambda_{\mathcal{C}_A}(w)\geq 0 $. So, $ \mu_{\mathcal{C}_A}(w) = 1 $ and $ \lambda_{\mathcal{C}_A}(w) = 0 $, for all $ w\in xyz $. It turns out that $ w\in A $. This shows that $ SAS\subseteq A $. Therefore, $ A $ is an interior hyperideal of $ S $.
Example 3.12. Let $ S = \{a, b, c, d\} $ be a set with the hyperoperation $ \circ $ on $ S $ defined by the following table:
° | a | b | c | d |
a | {a} | {a} | {a} | {a} |
b | {a} | {a} | {a} | {a} |
c | {a} | {a} | {a} | {a, b} |
d | {a} | {a} | {a, b} | {a, b, c} |
It follows that $ (S, \circ) $ is a semihypergroup, [40]. We see that $ A = \{a, c\} $ is an interior hyperideal of $ S $. After that, the FFS $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ of $ S $ defined by
$\mu_{\mathcal{A}}(x) = {1if x∈A,0otherwise, \;{\rm{and}}\;\lambda_{\mathcal{A}}(x) = {0if x∈A,1otherwise, $
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for all $ x\in S $. Applying Theorem 3.11, we have $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ is a FFInt of $ S $.
Proposition 3.13. Every FFH of a semihypergroup $ S $ is also an FFInt of $ S $.
Proof. Let $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ be an FFH of a semihypergroup $ S $. By Lemma 2.3, we have $ \mathcal{A}\circ\mathcal{A}\subseteq \mathcal{A}\circ\mathcal{S}\subseteq\mathcal{A} $ and $ \mathcal{S}\circ\mathcal{A}\circ\mathcal{S} = (\mathcal{S}\circ\mathcal{A})\circ\mathcal{S} \subseteq \mathcal{A}\circ\mathcal{S}\subseteq \mathcal{A} $. By Theorem 3.10, it follows that $ \mathcal{A} $ is an FFInt of $ S $.
Example 3.14. Let $ S = \{a, b, c, d\} $ such that $ (S, \circ) $ is a semihypergroup, as defined in Example 3.12. In the next step, we define an FFS $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ on $ S $ as follows:
![]() |
$ a $ | $ b $ | $ c $ | $ d $ |
$ \mu_{\mathcal{A}} $ | $ 0.9 $ | $ 0.6 $ | $ 0.8 $ | $ 0.5 $ |
$ \lambda_{\mathcal{A}} $ | $ 0.5 $ | $ 0.8 $ | $ 0.7 $ | $ 0.9 $ |
Upon careful inspection, we obtain that the FFS $ \mathcal{A} $ is an FFInt of $ S $. However, the FFInt $ \mathcal{A} $ of $ S $ is not a FFL of $ S $, because
$ {\inf\limits_{z\in d\circ c}}\mu_{\mathcal{A}}(z) = \mu_{\mathcal{A}}(b) < \mu_{\mathcal{A}}(c)\;{\rm{and}}\; {\sup\limits_{z\in d\circ c}}\lambda_{\mathcal{A}}(z) = \lambda_{\mathcal{A}}(b) > \lambda_{\mathcal{A}}(c)$. |
Furthermore, the FFInt $ \mathcal{A} $ of $ S $ is not an FFR of $ S $ either, since
$ {\inf\limits_{z\in c\circ d}}\mu_{\mathcal{A}}(z) = \mu_{\mathcal{A}}(b) < \mu_{\mathcal{A}}(c)\;{\rm{and}}\; {\sup\limits_{z\in c\circ d}}\lambda_{\mathcal{A}}(z) = \lambda_{\mathcal{A}}(b) > \lambda_{\mathcal{A}}(c)$. |
It can be concluded that the FFInt of $ S $ does not have to be an FFH of $ S $.
Theorem 3.15. In an intra-regular semihypergroup $ S $, every FFInt of $ S $ is also an FFH of $ S $.
Proof. Let $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ be an FFInt of $ S $, and let $ a, b\in S $. Then, there exist $ x, y\in S $ such that $ a\in xa^2y $. So, $ ab\subseteq (xa^2y)b = (xa)a(yb) $. Thus, for every $ z\in ab $, there exist $ u\in xa $ and $ v\in yb $ such that $ z\in uav $, which implies that $ \mu_{\mathcal{A}}(z)\geq \inf_{z\in uav}\mu_{\mathcal{A}}(z)\geq \mu_{\mathcal{A}}(a) $ and $ \lambda_{\mathcal{A}}(z)\leq \sup_{z\in uav}\lambda_{\mathcal{A}}(z)\leq \lambda_{\mathcal{A}}(a) $. We obtain that $ \inf_{z\in ab}\mu_{\mathcal{A}}(z)\geq \mu_{\mathcal{A}}(a) $ and $ \sup_{z\in ab}\lambda_{\mathcal{A}}(a) $. Hence, $ \mathcal{A} $ is an FFR of $ S $. Similarly, we can show that $ \mathcal{A} $ is an FFL of $ S $. Therefore, $ \mathcal{A} $ is an FFH of $ S $.
Theorem 3.16. Let $ S $ be a semihypergroup. Then the following results are equivalent:
(ⅰ) $ S $ is intra-regular;
(ⅱ) $ \mathcal{I}\cap\mathcal{G}\cap\mathcal{L}\subseteq \mathcal{L}\circ\mathcal{G}\circ\mathcal{I} $, for each FFL $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $, each FFInt $ \mathcal{I} = (\mu_{\mathcal{I}}, \lambda_{\mathcal{I}}) $ and each FFGB $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ of $ S $;
(ⅲ) $ \mathcal{I}\cap\mathcal{B}\cap\mathcal{L}\subseteq \mathcal{L}\circ\mathcal{B}\circ\mathcal{I} $, for each FFL $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $, each FFInt $ \mathcal{I} = (\mu_{\mathcal{I}}, \lambda_{\mathcal{I}}) $ and each FFB $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ of $ S $.
Proof. (ⅰ)$ \Rightarrow $(ⅱ) Assume that $ S $ is intra-regular. Let $ a\in S $. Then, there exist $ x, y\in S $ such that $ a\in xa^2y $, and so $ a\in (x^2a)a(yay) $. Thus, $ a\in waq $, for some $ w\in x^2a $ and $ q\in yay $, and then $ a\in pq, $ for some $ p\in wa $. So, we have
$ (μL∘μG∘μI)(a)=supa∈pq[min{(μL∘μG)(p),μI(q)}]=supa∈pq[min{supp∈wa[min{μL(w),μG(a)}],μI(q)}]≥min{min{infw∈x2aμL(w),μG(a)},infq∈yayμI(q)}≥min{min{μL(a),μG(a)},μI(a)}=min{μL(a),μG(a),μI(a)}=(μL∩μG∩μI)(a), $
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and
$ (λL∘λG∘λI)(a)=infa∈pq[max{(λL∘λG)(p),λI(q)}]=infa∈pq[max{infp∈wa[max{λL(w),λG(a)}],λI(q)}]≤max{max{supw∈x2aλL(w),λG(a)},supq∈yayλI(q)}≤max{max{λL(a),λG(a)},λI(a)}=max{λL(a),λG(a),λI(a)}=(λL∪λG∪λI)(a). $
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Therefore, $ \mathcal{I}\cap\mathcal{G}\cap\mathcal{L}\subseteq \mathcal{L}\circ\mathcal{G}\circ\mathcal{I} $.
(ⅱ)$ \Rightarrow $(ⅲ) Since every FFB of $ S $ is an FFGB of $ S $, it follows that (ⅲ) is obtained.
(ⅲ)$ \Rightarrow $(ⅰ) Let $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ and $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ be an FFL and an FFR of $ S $, respectively. Then, $ \mathcal{R} $ is also an FFB of $ S $. By assumption, we have $ \mathcal{L}\cap\mathcal{R} = \mathcal{S}\cap\mathcal{L}\cap\mathcal{R}\subseteq \mathcal{L}\circ(\mathcal{R}\circ\mathcal{S}) \subseteq \mathcal{L}\circ\mathcal{R} $. Consequently, $ S $ is intra-regular by Theorem 3.1.
Theorem 3.17. Let $ S $ be a semihypergroup. Then the following results are equivalent:
(ⅰ) $ S $ is intra-regular;
(ⅱ) $ \mathcal{I}\cap\mathcal{G}\cap\mathcal{R}\subseteq \mathcal{I}\circ\mathcal{G}\circ\mathcal{R} $, for each FFR $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $, each FFInt $ \mathcal{I} = (\mu_{\mathcal{I}}, \lambda_{\mathcal{I}}) $ and each FFGB $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ of $ S $;
(ⅲ) $ \mathcal{I}\cap\mathcal{B}\cap\mathcal{R}\subseteq \mathcal{I}\circ\mathcal{B}\circ\mathcal{R} $, for each FFR $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $, each FFInt $ \mathcal{I} = (\mu_{\mathcal{I}}, \lambda_{\mathcal{I}}) $ and each FFB $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ of $ S $.
Proof. (ⅰ)$ \Rightarrow $(ⅱ) Assume that $ S $ is intra-regular. Let $ a\in S $. Then, there exist $ x, y\in S $ such that $ a\in xa^2y $. This implies that $ a\in (xax)a(ay^2) $. Thus, $ a\in paw $, for some $ p\in xax $ and $ w\in ay^2 $, and so $ a\in pq $, for some $ q\in aw $. So, we have
$ (μI∘μG∘μR)(a)=supa∈pq[min{μI(p),(μG∘μR)(q)}]=supa∈pq[min{μI(p),supq∈aw[min{μG(a),μR(w)}]}]≥min{infp∈xaxμI(p),min{μG(a),infw∈ay2μR(w)}}≥min{μI(a),min{μG(a),μR(a)}}=min{μI(a),μG(a),μR(a)}=(μI∩μG∩μR)(a), $
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and
$ (λI∘λG∘λR)(a)=infa∈pq[max{λI(p),(λG∘λR)(q)}]=infa∈pq[max{λI(p),infq∈aw[max{λG(a),λR(w)}]}]≤max{supp∈xaxλI(p),max{λG(a),supw∈ay2λR(w)}}≤max{λI(a),max{λG(a),λR(a)}}=max{λI(a),λG(a),λR(a)}=(λI∪λG∪λR)(a). $
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It turns out that $ \mathcal{I}\cap\mathcal{G}\cap\mathcal{R}\subseteq \mathcal{I}\circ\mathcal{G}\circ\mathcal{R} $.
(ⅱ)$ \Rightarrow $(ⅲ) It is obvious.
(ⅲ)$ \Rightarrow $(ⅰ) Let $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ and $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ be an FFL and an FFR of $ S $, respectively. Then, $ \mathcal{R} $ is also an FFB of $ \mathcal{S} $. By assumption, we have $ \mathcal{L}\cap\mathcal{R} = \mathcal{S}\cap\mathcal{L}\cap\mathcal{R}\subseteq (\mathcal{S}\circ\mathcal{L})\circ\mathcal{R} \subseteq \mathcal{L}\circ\mathcal{R} $. By Theorem 3.1, we obtain that $ S $ is intra-regular.
In this section, we characterize both regular and intra-regular semihypergroups in terms of different types of Fermatean fuzzy hyperideals of semihypergroups.
Lemma 4.1. [35] Let $ S $ be a semihypergroup. Then, $ S $ is both regular and intra-regular if and only if $ B = BB $, for every bi-hyperideal $ B $ of $ S $.
Theorem 4.2. Let $ S $ be a semihypergroup. Then the following statements are equivalent:
(ⅰ) $ S $ is both regular and intra-regular;
(ⅱ) $ \mathcal{B} = \mathcal{B}\circ\mathcal{B} $, for any FFB $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ of $ S $;
(ⅲ) $ \mathcal{G}\cap\mathcal{H}\subseteq \mathcal{G}\circ\mathcal{H} $, for all FFGBs $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ and $ \mathcal{H} = (\mu_{\mathcal{H}}, \lambda_{\mathcal{H}}) $ of $ S $;
(ⅳ) $ \mathcal{A}\cap\mathcal{B}\subseteq \mathcal{A}\circ\mathcal{B} $, for all FFBs $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ and $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ of $ S $.
Proof. (ⅰ)$ \Rightarrow $(ⅲ) Let $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ and $ \mathcal{H} = (\mu_{\mathcal{H}}, \lambda_{\mathcal{H}}) $ be FFGBs of $ S $. By assumption, there exist $ x, y, z\in S $ such that $ a\in axa $ and $ a\in ya^2z $. Also, $ a\in (axya)(azxa) $, which implies that $ a\in pq $, for some $ p\in axya $ and $ q\in azxa $. Thus, we have
$ (μG∘μH)(a)=supa∈pq[min{μG(p),μH(q)}]≥min{infp∈axyaμG(p),infq∈azxaμH(q)}≥min{min{μG(a),μG(a)},min{μH(a),μH(a)}}=min{μG(a),μH(a)}=(μG∩μH)(a), $
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and
$ (λG∘λH)(a)=infa∈pq[max{λG(p),λH(q)}]≤max{supp∈axyaλG(p),supq∈azxaλH(q)}≤max{max{λG(a),λG(a)},max{λH(a),λH(a)}}=max{λG(a),λH(a)}=(λG∪λH)(a). $
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Therefore, $ \mathcal{G}\cap\mathcal{H}\subseteq \mathcal{G}\circ\mathcal{H} $.
(ⅲ)$ \Rightarrow $(ⅳ) Since every FFB is also an FFGB of $ S $, it follows that (iv) holds.
(ⅳ)$ \Rightarrow $(ⅱ) Let $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ be any FFB of $ S $. By the hypothesis, we have $ \mathcal{B} = \mathcal{B}\cap\mathcal{B}\subseteq \mathcal{B}\circ\mathcal{B} $. Otherwise, $ \mathcal{B}\circ\mathcal{B}\subseteq \mathcal{B} $ always. Hence, $ \mathcal{B} = \mathcal{B}\circ\mathcal{B} $.
(ⅱ)$ \Rightarrow $(ⅰ) Let $ B $ be any bi-hyperideal of $ S $. By Lemma 2.4, we have $ \mathcal{C}_B = (\mu_{\mathcal{C}_B}, \lambda_{\mathcal{C}_B}) $ is an FFB of $ S $. By the given assumption and Lemma 2.1, it follows that $ \mathcal{C}_B = \mathcal{C}_B\circ\mathcal{C}_B = \mathcal{C}_{BB} $. For every $ a\in B $, we have $ \mu_{\mathcal{C}_{BB}}(a) = \mu_{\mathcal{C}_{B}}(a) = 1 $. This means that $ a\in BB $. It turns out that $ B\subseteq BB $. On the other hand, $ BB\subseteq B $. Hence, $ B = BB $. By Lemma 4.1, we obtain that $ S $ is both regular and intra-regular.
The next theorem follows by Theorem 4.2.
Theorem 4.3. The following properties are equivalent in a semihypergroup $ S $:
(ⅰ) $ S $ is both regular and intra-regular;
(ⅱ) $ \mathcal{B}\cap\mathcal{G}\subseteq \mathcal{B}\circ\mathcal{G} $, for each FFB $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ and each FFGB $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ of $ S $;
(ⅲ) $ \mathcal{B}\cap\mathcal{G}\subseteq \mathcal{G}\circ\mathcal{B} $, for each FFB $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ and each FFGB $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ of $ S $.
Moreover, the following corollary obtained by Theorems 4.2 and 4.3.
Corollary 4.4. For a semihypergroup $ S $, the following conditions are equivalent:
(ⅰ) $ S $ is both regular and intra-regular;
(ⅱ) $ \mathcal{G}\cap\mathcal{H}\subseteq (\mathcal{G}\circ\mathcal{H})\cap(\mathcal{H}\circ\mathcal{G}) $, for all FFGBs $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ and $ \mathcal{H} = (\mu_{\mathcal{H}}, \lambda_{\mathcal{H}}) $ of $ S $;
(ⅲ) $ \mathcal{A}\cap\mathcal{B}\subseteq (\mathcal{A}\circ\mathcal{B})\cap(\mathcal{B}\circ\mathcal{A}) $, for all FFBs $ \mathcal{A} = (\mu_{\mathcal{A}}, \lambda_{\mathcal{A}}) $ and $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ of $ S $;
(ⅳ) $ \mathcal{B}\cap\mathcal{G}\subseteq (\mathcal{B}\circ\mathcal{G})\cap(\mathcal{G}\circ\mathcal{B}) $, for any FFB $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ and any FFGB $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ of $ S $.
By Lemma 2.6 and Theorem 3.1, we receive the following theorem.
Theorem 4.5. Let $ S $ be a semihypergroup. Then, $ S $ is both regular and intra-regular if and only if $ \mathcal{L}\cap\mathcal{R}\subseteq (\mathcal{L}\circ\mathcal{R})\cap(\mathcal{R}\circ\mathcal{L}) $, for every FFL $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ and every FFR $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ of $ S $.
The following theorem can be proved by Corollary 4.4 and Theorem 4.5.
Theorem 4.6. In a semihypergroup $ S $, the following statements are equivalent:
(ⅰ) $ S $ is both regular and intra-regular;
(ⅱ) $ \mathcal{G}\cap\mathcal{L}\subseteq (\mathcal{G}\circ\mathcal{L})\cap(\mathcal{L}\circ\mathcal{G}) $, for any FFL $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ and any FFGB $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ of $ S $;
(ⅲ) $ \mathcal{B}\cap\mathcal{L}\subseteq (\mathcal{B}\circ\mathcal{L})\cap(\mathcal{L}\circ\mathcal{B}) $, for any FFL $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ and any FFB $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ of $ S $;
(ⅳ) $ \mathcal{R}\cap\mathcal{G}\subseteq (\mathcal{G}\circ\mathcal{R})\cap(\mathcal{R}\circ\mathcal{G}) $, for every FFR $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ and any FFGB $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ of $ S $;
(ⅴ) $ \mathcal{R}\cap\mathcal{B}\subseteq (\mathcal{B}\circ\mathcal{R})\cap(\mathcal{R}\circ\mathcal{B}) $, for every FFR $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ and any FFB $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ of $ S $.
Theorem 4.7. The following properties are equivalent on a semihypergroup $ S $:
(ⅰ) $ S $ is both regular and intra-regular;
(ⅱ) $ \mathcal{L}\cap\mathcal{G}\subseteq \mathcal{G}\circ\mathcal{L}\circ\mathcal{G} $, for each FFL $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ and each FFGB $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ of $ S $;
(ⅲ) $ \mathcal{L}\cap\mathcal{B}\subseteq \mathcal{B}\circ\mathcal{L}\circ\mathcal{B} $, for each FFL $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ and each FFB $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ of $ S $;
(ⅳ) $ \mathcal{R}\cap\mathcal{G}\subseteq \mathcal{G}\circ\mathcal{R}\circ\mathcal{G} $, for each FFR $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ and each FFGB $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ of $ S $;
(ⅴ) $ \mathcal{R}\cap\mathcal{B}\subseteq \mathcal{B}\circ\mathcal{R}\circ\mathcal{B} $, for each FFR $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ and each FFB $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ of $ S $.
Proof. (ⅰ)$ \Rightarrow $(ⅱ) Let $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $ and $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ be an FFL and an FFGB of $ S $, respectively. Let $ a\in S $. Then, there exist $ x, y, z\in S $ such that $ a\in axa $ and $ a\in ya^2z $. This implies that $ a\in (axya)(azxya)(azxa) $; that is, $ a\in pq $, for some $ p\in axya $ and $ q\in uv $, where $ u\in azxya $ and $ v\in azxa $. Thus, we have
$ (μG∘μL∘μG)(a)=supa∈pq[min{μG(p),(μL∘μG)(q)}]=supa∈pq[min{μG(p),supq∈uv[min{μL(u),μG(v)}]}]≥min{infp∈axyaμG(p),min{infu∈azxyaμL(u),infv∈azxaμG(v)}}≥min{min{μG(a),μG(a)},min{μL(a),min{μG(a),μG(a)}}}=min{μL(a),μG(a)}=(μL∩μG)(a), $
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and
$ (λG∘λL∘λG)(a)=infa∈pq[max{λG(p),(λL∘λG)(q)}]=infa∈pq[max{λG(p),infq∈uv[max{λL(u),λG(v)}]}]≤max{supp∈axyaλG(p),max{supu∈azxyaλL(u),supv∈azxaλG(v)}}≤max{max{λG(a),λG(a)},max{λL(a),max{λG(a),λG(a)}}}=max{λL(a),λG(a)}=(λL∪λG)(a). $
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We obtain that $ \mathcal{L}\cap\mathcal{G}\subseteq \mathcal{G}\circ\mathcal{L}\circ\mathcal{G} $.
(ⅱ)$ \Rightarrow $(ⅲ) It follows by the fact that every FFB is also an FFGB of $ S $.
(ⅲ)$ \Rightarrow $(ⅰ) Let $ a\in S $. It is easy to verify that $ a\cup Sa $ and $ a\cup aa\cup aSa $ are a left hyperideal and a bi-hyperideal of $ S $ with containing $ a $, respectively. Then, $ \mathcal{C}_{a\cup Sa} $ and $ \mathcal{C}_{a\cup aa\cup aSa} $ are an FFL and an FFB of $ S $, respectively. By the given hypothesis and Lemma 2.1, we obtain:
$ C(a∪Sa)∩(a∪aa∪aSa)=Ca∪Sa∩Ca∪aa∪aSa⊆Ca∪aa∪aSa∘Ca∪Sa∘Ca∪aa∪aSa=C(a∪aa∪aSa)(a∪Sa)(a∪aa∪aSa). $
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This means that $ \mu_{\mathcal{C}_{(a\cup aa\cup aSa)(a\cup Sa)(a\cup aa\cup aSa)}}(a) \geq \mu_{\mathcal{C}_{(a\cup Sa)\cap(a\cup aa\cup aSa)}}(a) = 1 $. Also, $ a\in (a\cup aa\cup aSa)(a\cup Sa)(a\cup aa\cup aSa) $. It turns out that $ a\in (aSa)\cap(Sa^2S) $. Consequently, $ S $ is both regular and intra-regular.
Similarly, we can prove that (ⅰ)$ \Rightarrow $(ⅳ)$ \Rightarrow $(ⅴ)$ \Rightarrow $(ⅰ) obtain.
Theorem 4.8. Let $ S $ be a semihypergroup. Then the following statements are equivalent:
(ⅰ) $ S $ is both regular and intra-regular;
(ⅱ) $ \mathcal{L}\cap\mathcal{R}\cap\mathcal{G}\subseteq \mathcal{G}\circ\mathcal{R}\circ\mathcal{L} $, for every FFL $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $, every FFR $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ and every FFGB $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ of $ S $;
(ⅲ) $ \mathcal{L}\cap\mathcal{R}\cap\mathcal{B}\subseteq \mathcal{B}\circ\mathcal{R}\circ\mathcal{L} $, for every FFL $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $, every FFR $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ and every FFB $ \mathcal{B} = (\mu_{\mathcal{B}}, \lambda_{\mathcal{B}}) $ of $ S $.
Proof. (ⅰ)$ \Rightarrow $(ⅱ) Let $ \mathcal{L} = (\mu_{\mathcal{L}}, \lambda_{\mathcal{L}}) $, $ \mathcal{R} = (\mu_{\mathcal{R}}, \lambda_{\mathcal{R}}) $ and $ \mathcal{G} = (\mu_{\mathcal{G}}, \lambda_{\mathcal{G}}) $ be an FFL, FFR, and FFGB of $ S $, respectively. Then, for any $ a\in S $, there exist $ x, y, z\in S $ such that $ a\in axa $ and $ a\in ya^2z $. So, $ a\in (axya)(az)(xa) $. Also, $ a\in pq $, for some $ p\in axya $ and $ q\in uv $, where $ u\in az $ and $ v\in xa $. Thus, we have
$ (μG∘μR∘μL)(a)=supa∈pq[min{μG(p),(μR∘μL)(q)}]=supa∈pq[min{μG(p),supq∈uv[min{μR(u),μL(v)}]}]≥min{infp∈axyaμG(p),min{infu∈azμR(u),infv∈xaμL(v)}}≥min{min{μG(a),μG(a)},min{μR(a),μL(a)}}=min{μG(a),μR(a),μL(a)}=(μG∩μR∩μL)(a), $
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and
$ (λG∘λR∘λL)(a)=infa∈pq[max{λG(p),(λR∘λL)(q)}]=infa∈pq[max{λG(p),infq∈uv[max{λR(u),λL(v)}]}]≤max{supp∈axyaλG(p),max{supu∈azλR(u),supv∈xaλL(v)}}≤max{max{λG(a),λG(a)},max{λR(a),λL(a)}}=max{λG(a),λR(a),λL(a)}=(λG∪λR∪λL)(a). $
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It follows that $ \mathcal{L}\cap\mathcal{R}\cap\mathcal{G}\subseteq \mathcal{G}\circ\mathcal{R}\circ\mathcal{L} $.
(ⅱ)$ \Rightarrow $(ⅲ) It is obvious.
(ⅲ)$ \Rightarrow $(ⅰ) Let $ s\in S $. It is not difficult to show that the sets $ a\cup Sa $, $ a\cup aS $, and $ a\cup aa\cup aSa $ are a left hyperideal, a right hyperideal, and a bi-hyperideal of $ S $ with containing $ a $, respectively. By Lemma 2.4, we have $ \mathcal{C}_{a\cup Sa} $, $ \mathcal{C}_{a\cup aS} $, and $ \mathcal{C}_{a\cup aa\cup aSa} $ are an FFL, an FFR, and an FFB of $ S $, respectively. Using the assumption and Lemma 2.1, we have
$ C(a∪Sa)∩(a∪aS)∩(a∪aa∪aSa)=Ca∪Sa∩Ca∪aS∩Ca∪aa∪aSa⊆Ca∪aa∪aSa∘Ca∪aS∘Ca∪Sa=C(a∪aa∪aSa)(a∪aS)(a∪Sa). $
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It turns out that $ \mu_{\mathcal{C}_{(a\cup aa\cup aSa)(a\cup aS)(a\cup Sa)}}(a) \geq \mu_{\mathcal{C}_{(a\cup Sa)\cap(a\cup aS)\cap(a\cup aa\cup aSa)}}(a) = 1 $; that is, $ a\in (a\cup aa\cup aSa)(a\cup aS)(a\cup Sa) $. Thus, $ a\in (aSa)\cap(Sa^2S) $. Therefore, $ S $ is both regular and intra-regular.
In 2023, Nakkhasen [28] applied the concept of Fermatean fuzzy sets to characterize the class of regular semihypergroups. In this research, we discussed the characterizations of intra-regular semihypergroups using the properties of Fermatean fuzzy left hyperideals, Fermatean fuzzy right hyperideals, Fermatean fuzzy generalized bi-hyperideals, and Fermatean fuzzy bi-hyperideals of semihypergroups, which are shown in Section 3. In addition, we introduced the concept of Fermatean fuzzy interior hyperideals of semihypergroups and used this concept to characterize intra-regular semihypergroups and proved that Fermatean fuzzy interior hyperideals and Fermatean fuzzy hyperideals coincide in intra-regular semihypergroups. Furthermore, in Section 4, the characterizations of both regular and intra-regular semihypergroups by many types of their Fermatean fuzzy hyperideals are presented. In our next paper, we will investigate the characterization of weakly regular semihypergroups using different types of Fermatean fuzzy hyperideals of semihypergroups. Additionally, we will use the attributes of Fermatean fuzzy sets to describe various regularities (e.g., left regular, right regular, and completely regular) in semihypergroups.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
Warud Nakkhasen: conceptualization, investigation, original draft preparation, writing-review & editing, supervision; Teerapan Jodnok: writing-review & editing, supervision; Ronnason Chinram: writing-review & editing, supervision. All authors have read and approved the final version of the manuscript for publication.
This research project was financially supported by Thailand Science Research and Innovation (TSRI).
The authors declare no conflict of interest.
[1] |
Peacock SJ, de Silva I, Lowy FD (2001) What determines nasal carriage of Staphylococcus aureus? Trends Microbiol 9: 605-610. doi: 10.1016/S0966-842X(01)02254-5
![]() |
[2] |
Foster TJ. The Staphylococcus aureus “superbug” (2004) J Clin Invest 114: 1693-1696. doi: 10.1172/JCI200423825
![]() |
[3] |
Wertheim HF, Melles DC, Vos MC, et al. (2005) The role of nasal carriage in Staphylococcus aureus infections. Lancet Infect Dis 5: 751-762. doi: 10.1016/S1473-3099(05)70295-4
![]() |
[4] |
Eriksen NH, Espersen F, Rosdahl VT, et al. (1995) Carriage of Staphylococcus aureus among 104 healthy persons during a 19-month period. Epidemiol Infect 115: 51-60. doi: 10.1017/S0950268800058118
![]() |
[5] |
Weidenmaier C, Goerke C, Wolz C (2012) Staphylococcus aureus determinants for nasal colonization. Trends Microbiol 20: 243-250. doi: 10.1016/j.tim.2012.03.004
![]() |
[6] |
Foster TJ (2009) Colonization and infection of the human host by staphylococci: Adhesion, survival and immune evasion. Vet Dermatol 20: 456-470. doi: 10.1111/j.1365-3164.2009.00825.x
![]() |
[7] |
O'Brien LM, Walsh EJ, Massey RC, et al. (2002) Staphylococcus aureus clumping factor b (Clfb) promotes adherence to human type I cytokeratin 10: Implications for nasal colonization. Cell Microbiol 4: 759-770. doi: 10.1046/j.1462-5822.2002.00231.x
![]() |
[8] |
Hauck CR, Ohlsen K (2006) Sticky connections: Extracellular matrix protein recognition and integrin-mediated cellular invasion by Staphylococcus aureus. Curr Opin Microbiol 9: 5-11. doi: 10.1016/j.mib.2005.12.002
![]() |
[9] | Mongodin E, Bajolet O, Cutrona J, et al. (2002) Fibronectin-binding proteins of Staphylococcus aureus are involved in adherence to human airway epithelium. Infect Immun 70: 620-630. |
[10] |
Roche FM, Downer R, Keane F, et al. (2004) The N-terminal a domain of fibronectin-binding proteins A and B promotes adhesion of Staphylococcus aureus to elastin.J Biol Chem 279: 38433-38440. doi: 10.1074/jbc.M402122200
![]() |
[11] |
Weidenmaier C, Kokai-Kun JF, Kristian SA, et al. (2004) Role of teichoic acids in Staphylococcus aureus nasal colonization, a major risk factor in nosocomial infections. Nat Med 10: 243-245. doi: 10.1038/nm991
![]() |
[12] |
Patti JM, Allen BL, McGavin MJ, et al. (1994) MSCRAMM-mediated adherence of microorganisms to host tissues. Annu Rev Microbiol 48: 585-617. doi: 10.1146/annurev.mi.48.100194.003101
![]() |
[13] |
Clarke SR, Foster SJ (2006) Surface adhesins of Staphylococcus aureus. Adv Microb Physiol 51: 187-224. doi: 10.1016/S0065-2911(06)51004-5
![]() |
[14] |
van Belkum A, Melles DC, Nouwen J, et al. (2009) Co-evolutionary aspects of human colonisation and infection by Staphylococcus aureus. Infect Genet Evol 9: 32-47. doi: 10.1016/j.meegid.2008.09.012
![]() |
[15] |
Armstrong-Esther CA (1976) Carriage patterns of Staphylococcus aureus in a healthy non-hospital population of adults and children. Ann Hum Biol 3: 221-227. doi: 10.1080/03014467600001381
![]() |
[16] |
Bassetti S, Dunagana DP, D'Agostino RB, et al. (2001) Nasal carriage of Staphylococcus aureus among patients receiving allergen-injection immunotherapy: Associated factors and quantitative nasal cultures. Infect Contr Hosp Epidem 22: 741-745. doi: 10.1086/501857
![]() |
[17] |
Miller M, Cespedes C, Bhat M, et al. (2007) Incidence and persistence of Staphylococcus aureus nasal colonization in a community sample of HIV-infected and -uninfected drug users. Clin Infect Dis 45: 343-346. doi: 10.1086/519429
![]() |
[18] |
von Eiff C, Becker K, Machka K, et al. (2001) Nasal carriage as a source of Staphylococcus aureus bacteremia. Study group. N Engl J Med 344: 11-16. doi: 10.1056/NEJM200101043440102
![]() |
[19] |
Fritz SA, Tiemann KM, Hogan PG, et al. (2013) A serologic correlate of protective immunity against community-onset Staphylococcus aureus infection. Clin Infect Dis 56: 1554-1561. doi: 10.1093/cid/cit123
![]() |
[20] |
Peacock SJ, Moore CE, Justice A, et al. (2002) Virulent combinations of adhesin and toxin genes in natural populations of Staphylococcus aureus. Infect Immun 70: 4987-4996. doi: 10.1128/IAI.70.9.4987-4996.2002
![]() |
[21] |
Melles DC, Gorkink RF, Boelens HA, et al. (2004) Natural population dynamics and expansion of pathogenic clones of Staphylococcus aureus. J Clin Invest 114: 1732-1740. doi: 10.1172/JCI200423083
![]() |
[22] | Kluytmans J, van Belkum A, Verbrugh H (1997) Nasal carriage of Staphylococcus aureus: Epidemiology, underlying mechanisms, and associated risks. Clin Microbiol Rev 10: 505-520. |
[23] |
Lee MH, Arrecubieta C, Martin FJ, et al. (2010) A postinfluenza model of Staphylococcus aureus pneumonia. J Infect Dis 201: 508-515. doi: 10.1086/650204
![]() |
[24] |
Handler MZ, Schwartz RA (2014) Staphylococcal scalded skin syndrome: Diagnosis and management in children and adults. J Eur Acad Dermatol Venereol 28: 1418-1423. doi: 10.1111/jdv.12541
![]() |
[25] | Ibler KS, Kromann CB (2014) Recurrent furunculosis - challenges and management: A review. Clin Cosm Invest Dermatol 7: 59-64. |
[26] |
Malik Z, Roscioli E, Murphy J, et al. (2015) Staphylococcus aureus impairs the airway epithelial barrier in vitro. Int Forum Allergy Rhinol 5: 551-556. doi: 10.1002/alr.21517
![]() |
[27] |
Barbier F, Andremont A, Wolff M, et al. (2013) Hospital-acquired pneumonia and ventilator-associated pneumonia: Recent advances in epidemiology and management. Curr Opin Pulm Med 19: 216-228. doi: 10.1097/MCP.0b013e32835f27be
![]() |
[28] |
Cohen J (2002) The immunopathogenesis of sepsis. Nature 420: 885-891. doi: 10.1038/nature01326
![]() |
[29] |
Fournier B, Philpott DJ (2005) Recognition of Staphylococcus aureus by the innate immune system. Clin Microbiol Rev 18: 521-540. doi: 10.1128/CMR.18.3.521-540.2005
![]() |
[30] |
Foster TJ (2005) Immune evasion by staphylococci. Nat Rev Microbiol 3: 948-958. doi: 10.1038/nrmicro1289
![]() |
[31] |
Enright MC, Robinson DA, Randle G, et al. (2002) The evolutionary history of methicillin-resistant Staphylococcus aureus (MRSA). Proc Nat Acad Sci USA 99: 7687-7692. doi: 10.1073/pnas.122108599
![]() |
[32] |
Deresinski S (2005) Methicillin-resistant Staphylococcus aureus: An evolutionary, epidemiologic, and therapeutic odyssey. Clin Infect Dis 40: 562-573. doi: 10.1086/427701
![]() |
[33] |
Kahl BC (2010) Impact of Staphylococcus aureus on the pathogenesis of chronic cystic fibrosis lung disease. Int J Med Microbiol 300: 514-519. doi: 10.1016/j.ijmm.2010.08.002
![]() |
[34] |
Ganesan S, Comstock AT, Sajjan US (2013) Barrier function of airway tract epithelium. Tiss Barriers 1: e24997. doi: 10.4161/tisb.24997
![]() |
[35] |
Knowles MR, Boucher RC (2002) Mucus clearance as a primary innate defense mechanism for mammalian airways. J Clin Invest 109: 571-577. doi: 10.1172/JCI0215217
![]() |
[36] |
Button B, Cai LH, Ehre C, et al. (2012) A periciliary brush promotes the lung health by separating the mucus layer from airway epithelia. Science 337: 937-941. doi: 10.1126/science.1223012
![]() |
[37] |
Schleimer RP, Kato A, Kern R, et al. (2007) Epithelium: At the interface of innate and adaptive immune responses. J Allergy Clin Immunol 120: 1279-1284. doi: 10.1016/j.jaci.2007.08.046
![]() |
[38] |
Kato A, Schleimer RP (2007) Beyond inflammation: Airway epithelial cells are at the interface of innate and adaptive immunity. Curr Opin Immunol 19: 711-720. doi: 10.1016/j.coi.2007.08.004
![]() |
[39] |
Diamond G, Legarda D, Ryan LK (2000) The innate immune response of the respiratory epithelium. Immunol Rev 173: 27-38. doi: 10.1034/j.1600-065X.2000.917304.x
![]() |
[40] |
Bals R, Hiemstra PS (2004) Innate immunity in the lung: How epithelial cells fight against respiratory pathogens. Eur Resp J 23: 327-333. doi: 10.1183/09031936.03.00098803
![]() |
[41] |
Zaas AK, Schwartz DA (2005) Innate immunity and the lung: Defense at the interface between host and environment. Trends Cardiovasc Med 15: 195-202. doi: 10.1016/j.tcm.2005.07.001
![]() |
[42] |
Parker D, Prince A (2011) Innate immunity in the respiratory epithelium. Am J Respir Cell Mol Biol 45: 189-201. doi: 10.1165/rcmb.2011-0011RT
![]() |
[43] |
Evans SE, Xu Y, Tuvim MJ, et al. (2010) Inducible innate resistance of lung epithelium to infection. Annu Rev Physiol 72: 413-435. doi: 10.1146/annurev-physiol-021909-135909
![]() |
[44] |
Ausubel FM (2005) Are innate immune signaling pathways in plants and animals conserved? Nat Immunol 6: 973-979. doi: 10.1038/ni1253
![]() |
[45] |
Sukhithasri V, Nisha N, Biswas L, et al. (2013) Innate immune recognition of microbial cell wall components and microbial strategies to evade such recognitions. Microbiol Res 168: 396-406. doi: 10.1016/j.micres.2013.02.005
![]() |
[46] |
Garcia-Vallejo JJ, van Kooyk Y (2009) Endogenous ligands for C-type lectin receptors: The true regulators of immune homeostasis. Immunol Rev 230: 22-37. doi: 10.1111/j.1600-065X.2009.00786.x
![]() |
[47] |
Kawai T, Akira S (2007) TLR signaling. Sem Immunol 19: 24-32. doi: 10.1016/j.smim.2006.12.004
![]() |
[48] |
Beutler B, Jiang Z, Georgel P, et al. (2006) Genetic analysis of host resistance: Toll-like receptor signaling and immunity at large. Annu Rev Immunol 24: 353-389. doi: 10.1146/annurev.immunol.24.021605.090552
![]() |
[49] |
Bubeck Wardenburg J, Williams WA, Missiakas D (2006) Host defenses against Staphylococcus aureus infection require recognition of bacterial lipoproteins. Proc Nat Acad Sci USA 103: 13831-13836. doi: 10.1073/pnas.0603072103
![]() |
[50] |
Strober W, Murray PJ, Kitani A, et al. (2006) Signalling pathways and molecular interactions of NOD1 and NOD2. Nat Rev Immunol 6: 9-20. doi: 10.1038/nri1747
![]() |
[51] |
Ginsburg I (2002) Role of lipoteichoic acid in infection and inflammation. Lancet Infect Dis 2: 171-179. doi: 10.1016/S1473-3099(02)00226-8
![]() |
[52] |
Schwandner R, Dziarski R, Wesche H, et al. (1999) Peptidoglycan- and lipoteichoic acid-induced cell activation is mediated by Toll-like receptor 2. J Biol Chem 274: 17406-17409. doi: 10.1074/jbc.274.25.17406
![]() |
[53] |
Beisswenger C, Coyne CB, Shchepetov M, et al. (2007) Role of p38 MAP kinase and transforming growth factor-beta signaling in transepithelial migration of invasive bacterial pathogens. J Biol Chem 282: 28700-28708. doi: 10.1074/jbc.M703576200
![]() |
[54] |
Soong G, Reddy B, Sokol S, et al. (2004) TLR2 is mobilized into an apical lipid raft receptor complex to signal infection in airway epithelial cells. J Clin Invest 113: 1482-1489. doi: 10.1172/JCI200420773
![]() |
[55] | Inohara N, Ogura Y, Fontalba A, et al. (2003) Host recognition of bacterial muramyl dipeptide mediated through NOD2. Implications for Crohn's disease. J Biol Chem 278: 5509-5512. |
[56] |
Gomez MI, Lee A, Reddy B, et al. (2004) Staphylococcus aureus protein a induces airway epithelial inflammatory responses by activating TNFR1. Nat Med 10: 842-848. doi: 10.1038/nm1079
![]() |
[57] |
Gomez MI, Prince A (2008) Airway epithelial cell signaling in response to bacterial pathogens. Pediatr Pulmonol 43: 11-19. doi: 10.1002/ppul.20735
![]() |
[58] | Prince AS, Mizgerd JP, Wiener-Kronish J, et al. (2006) Cell signaling underlying the pathophysiology of pneumonia. Am J Physiol 291: L297-L300. |
[59] |
Cheon IS, Woo SS, Kang SS, et al. (2008) Peptidoglycan-mediated IL-8 expression in human alveolar type II epithelial cells requires lipid raft formation and MAPK activation. Mol Immunol 45: 1665-1673. doi: 10.1016/j.molimm.2007.10.001
![]() |
[60] |
Liu L, Mul FP, Lutter R, et al. (1996) Transmigration of human neutrophils across airway epithelial cell monolayers is preferentially in the physiologic basolateral-to-apical direction. Am J Respir Cell Mol Biol 15: 771-780. doi: 10.1165/ajrcmb.15.6.8969272
![]() |
[61] |
Kishimoto T (2010) IL-6: From its discovery to clinical applications. Int Immunol 22: 347-352. doi: 10.1093/intimm/dxq030
![]() |
[62] |
Gauldie J, Richards C, Harnish D, et al. (1987) Interferon beta 2/B-cell stimulatory factor type 2 shares identity with monocyte-derived hepatocyte-stimulating factor and regulates the major acute phase protein response in liver cells. Proc Nat Acad Sci USA 84: 7251-7255. doi: 10.1073/pnas.84.20.7251
![]() |
[63] |
Ruminy P, Gangneux C, Claeyssens S, et al. (2001) Gene transcription in hepatocytes during the acute phase of a systemic inflammation: From transcription factors to target genes. Inflamm Res 50: 383-390. doi: 10.1007/PL00000260
![]() |
[64] |
Chavez-Galan L, Arenas-Del Angel MC, et al. (2009) Cell death mechanisms induced by cytotoxic lymphocytes. Cell Mol Immunol 6: 15-25. doi: 10.1038/cmi.2009.3
![]() |
[65] | Moreilhon C, Gras D, Hologne C, et al. (2005) Live Staphylococcus aureus and bacterial soluble factors induce different transcriptional responses in human airway cells. Physiol Genom 20: 244-255. |
[66] | Below S, Konkel A, Zeeck C, et al. (2009) Virulence factors of Staphylococcus aureus induce Erk-MAP kinase activation and c-fos expression in S9 and 16HBE14o- human airway epithelial cells. Am J Physiol 296: L470-L479. |
[67] | Räth S, Ziesemer S, Witte A, et al. (2013) S. aureus hemolysin A-induced IL-8 and IL-6 release from human airway epithelial cells is mediated by activation of p38- and Erk-MAP kinases and additional, cell-type specific signalling mechanisms. Cell Microbiol 15: 1253-1265. |
[68] | Rose F, Dahlem G, Guthmann B, et al. (2002) Mediator generation and signaling events in alveolar epithelial cells attacked by S. aureus alpha-toxin. Am J Physiol 282: L207-L214. |
[69] |
Bartlett AH, Foster TJ, Hayashida A, et al. (2008) Alpha-toxin facilitates the generation of CXC chemokine gradients and stimulates neutrophil homing in Staphylococcus aureus pneumonia. J Infect Dis 198: 1529-1535. doi: 10.1086/592758
![]() |
[70] |
Liang X, Ji Y (2007) Involvement of alpha5beta1-integrin and TNF-alpha in Staphylococcus aureus alpha-toxin-induced death of epithelial cells. Cell Microbiol 9: 1809-1821. doi: 10.1111/j.1462-5822.2007.00917.x
![]() |
[71] |
Ventura CL, Higdon R, Hohmann L, et al. (2008) Staphylococcus aureus elicits marked alterations in the airway proteome during early pneumonia. Infect Immun 76: 5862-5872. doi: 10.1128/IAI.00865-08
![]() |
[72] |
Ventura CL, Higdon R, Kolker E, et al. (2008) Host airway proteins interact with Staphylococcus aureus during early pneumonia. Infect Immun 76: 888-898. doi: 10.1128/IAI.01301-07
![]() |
[73] |
McElroy MC, Cain DJ, Tyrrell C, et al. (2002) Increased virulence of a fibronectin-binding protein mutant of Staphylococcus aureus in a rat model of pneumonia. Infect Immun 70: 3865-3873. doi: 10.1128/IAI.70.7.3865-3873.2002
![]() |
[74] |
Dziewanowska K, Carson AR, Patti JM, et al. (2000) Staphylococcal fibronectin binding protein interacts with heat shock protein 60 and integrins: Role in internalization by epithelial cells. Infect Immun 68: 6321-6328. doi: 10.1128/IAI.68.11.6321-6328.2000
![]() |
[75] | Surmann K, Michalik S, Hildebrandt P, et al. (2014) Comparative proteome analysis reveals conserved and specific adaptation patterns of Staphylococcus aureus after internalization by different types of human non-professional phagocytic host cells. Front Microbiol 5: 392. |
[76] |
Sinha B, Fraunholz M (2010) Staphylococcus aureus host cell invasion and post-invasion events. Int J Med Microbiol 300: 170-175. doi: 10.1016/j.ijmm.2009.08.019
![]() |
[77] |
Schnaith A, Kashkar H, Leggio SA, et al. (2007) Staphylococcus aureus subvert autophagy for induction of caspase-independent host cell death. J Biol Chem 282: 2695-2706. doi: 10.1074/jbc.M609784200
![]() |
[78] |
Schmidt F, Scharf SS, Hildebrandt P, et al. (2010) Time-resolved quantitative proteome profiling of host-pathogen interactions: The response of Staphylococcus aureus RN1HG to internalisation by human airway epithelial cells. Proteomics 10: 2801-2811. doi: 10.1002/pmic.201000045
![]() |
[79] |
Kahl BC, Goulian M, van Wamel W, et al. (2000) Staphylococcus aureus RN6390 replicates and induces apoptosis in a pulmonary epithelial cell line. Infect Immun 68: 5385-5392. doi: 10.1128/IAI.68.9.5385-5392.2000
![]() |
[80] |
Garzoni C, Kelley WL (2009) Staphylococcus aureus: New evidence for intracellular persistence. Trends Microbiol 17: 59-65. doi: 10.1016/j.tim.2008.11.005
![]() |
[81] |
Tuchscherr L, Medina E, Hussain M, et al. (2011) Staphylococcus aureus phenotype switching: An effective bacterial strategy to escape host immune response and establish a chronic infection. EMBO Mol Med 3: 129-141. doi: 10.1002/emmm.201000115
![]() |
[82] |
Phillips JR, Tripp TJ, Regelmann WE, et al. (2006) Staphylococcal alpha-toxin causes increased tracheal epithelial permeability. Pediatr Pulmonol 41: 1146-1152. doi: 10.1002/ppul.20501
![]() |
[83] |
Richter E, Harms M, Ventz K, et al. (2015) A multi-omics approach identifies key hubs associated with cell type-specific responses of airway epithelial cells to staphylococcal alpha-toxin. PLoS ONE 10: e0122089. doi: 10.1371/journal.pone.0122089
![]() |
[84] |
Hermann I, Räth S, Ziesemer S, et al. (2015) Staphylococcus aureus-hemolysin A disrupts cell-matrix adhesions in human airway epithelial cells. Am J Respir Cell Mol Biol 52: 14-24. doi: 10.1165/rcmb.2014-0082OC
![]() |
[85] |
Ratner AJ, Bryan R, Weber A, et al. (2001) Cystic fibrosis pathogens activate Ca2+-dependent mitogen-activated protein kinase signaling pathways in airway epithelial cells. J Biol Chem 276: 19267-19275. doi: 10.1074/jbc.M007703200
![]() |
[86] | Greene CM, Ramsay H, Wells RJ, et al. (2010) Inhibition of Toll-like receptor 2-mediated interleukin-8 production in cystic fibrosis airway epithelial cells via the alpha7-nicotinic acetylcholine receptor. Mediat Inflamm 2010: 423241. |
[87] |
Sibbald MJ, Ziebandt AK, Engelmann S, et al. (2006) Mapping the pathways to staphylococcal pathogenesis by comparative secretomics. Microbiol Mol Biol Rev 70: 755-788. doi: 10.1128/MMBR.00008-06
![]() |
[88] |
Kuroda M, Ohta T, Uchiyama I, et al. (2001) Whole genome sequencing of meticillin-resistant Staphylococcus aureus. Lancet 357: 1225-1240. doi: 10.1016/S0140-6736(00)04403-2
![]() |
[89] |
Gill SR, Fouts DE, Archer GL, et al. (2005) Insights on evolution of virulence and resistance from the complete genome analysis of an early methicillin-resistant Staphylococcus aureus strain and a biofilm-producing methicillin-resistant Staphylococcus epidermidis strain. J Bacteriol 187: 2426-2438. doi: 10.1128/JB.187.7.2426-2438.2005
![]() |
[90] |
Baba T, Takeuchi F, Kuroda M, et al. (2002) Genome and virulence determinants of high virulence community-acquired MRSA. Lancet 359: 1819-1827. doi: 10.1016/S0140-6736(02)08713-5
![]() |
[91] |
Diep BA, Carleton HA, Chang RF, et al. (2006) Roles of 34 virulence genes in the evolution of hospital- and community-associated strains of methicillin-resistant Staphylococcus aureus. J Infect Dis 193: 1495-1503. doi: 10.1086/503777
![]() |
[92] |
Goerke C, Wolz C (2010) Adaptation of Staphylococcus aureus to the cystic fibrosis lung. Int J Med Microbiol 300: 520-525. doi: 10.1016/j.ijmm.2010.08.003
![]() |
[93] |
Goerke C, Pantucek R, Holtfreter S, et al. (2009) Diversity of prophages in dominant Staphylococcus aureus clonal lineages. J Bacteriol 191: 3462-3468. doi: 10.1128/JB.01804-08
![]() |
[94] |
Novick RP (2003) Autoinduction and signal transduction in the regulation of staphylococcal virulence. Mol Microbiol 48: 1429-1249. doi: 10.1046/j.1365-2958.2003.03526.x
![]() |
[95] |
Cheung AL, Koomey JM, Butler CA, et al. (1992) Regulation of exoprotein expression in Staphylococcus aureus by a locus (sar) distinct from agr. Proc Nat Acad Sci USA 89: 6462-6466. doi: 10.1073/pnas.89.14.6462
![]() |
[96] |
Rogasch K, Rühmling V, Pane-Farre J, et al. (2006) Influence of the two-component system SaeRS on global gene expression in two different Staphylococcus aureus strains. J Bacteriol 188: 7742-7758. doi: 10.1128/JB.00555-06
![]() |
[97] |
Geisinger E, George EA, Muir TW, et al. (2008) Identification of ligand specificity determinants in AgrC, the Staphylococcus aureus quorum-sensing receptor. J Biol Chem 283: 8930-8938. doi: 10.1074/jbc.M710227200
![]() |
[98] |
Jensen RO, Winzer K, Clarke SR, et al. (2008) Differential recognition of Staphylococcus aureus quorum-sensing signals depends on both extracellular loops 1 and 2 of the transmembrane sensor AgrC. J Mol Biol 381: 300-309. doi: 10.1016/j.jmb.2008.06.018
![]() |
[99] |
Heyer G, Saba S, Adamo R, et al. (2002) Staphylococcus aureusagr and sarA functions are required for invasive infection but not inflammatory responses in the lung. Infect Immun 70: 127-133. doi: 10.1128/IAI.70.1.127-133.2002
![]() |
[100] |
Haslinger-Löffler B, Kahl BC, Grundmeier M, et al. (2005) Multiple virulence factors are required for Staphylococcus aureus-induced apoptosis in endothelial cells. Cell Microbiol 7: 1087-1097. doi: 10.1111/j.1462-5822.2005.00533.x
![]() |
[101] |
Jones RC, Deck J, Edmondson RD, et al. (2008) Relative quantitative comparisons of the extracellular protein profiles of Staphylococcus aureus UAMS-1 and its sarA, agr, and sarA agr regulatory mutants using one-dimensional polyacrylamide gel electrophoresis and nanocapillary liquid chromatography coupled with tandem mass spectrometry. J Bacteriol 190: 5265-5278. doi: 10.1128/JB.00383-08
![]() |
[102] |
Wright JS, Jin R, Novick RP (2005) Transient interference with staphylococcal quorum sensing blocks abscess formation. Proc Nat Acad Sci USA 102: 1691-1696. doi: 10.1073/pnas.0407661102
![]() |
[103] |
Feng Y, Chen CJ, Su LH, et al. (2008) Evolution and pathogenesis of Staphylococcus aureus: Lessons learned from genotyping and comparative genomics. FEMS Microbiol Rev 32: 23-37. doi: 10.1111/j.1574-6976.2007.00086.x
![]() |
[104] |
Hecker M, Becher D, Fuchs S, et al. (2010) A proteomic view of cell physiology and virulence of Staphylococcus aureus. Int J Med Microbiol 300: 76-87. doi: 10.1016/j.ijmm.2009.10.006
![]() |
[105] |
Becher D, Hempel K, Sievers S, et al. (2009) A proteomic view of an important human pathogen-towards the quantification of the entire Staphylococcus aureus proteome. PLoS ONE 4: e8176. doi: 10.1371/journal.pone.0008176
![]() |
[106] | Ziebandt A-K, Weber H, Rudolph J, et al. (2001) Extracellular proteins of Staphylococcus aureus and the role of SarA and σB. Proteomics 1: 480-493. |
[107] |
Ziebandt A-K, Becher D, Ohlsen K, et al. (2004) The influence of agr and σB in growth phase dependent regulation of virulence factors in Staphylococcus aureus. Proteomics 4: 3034-3047. doi: 10.1002/pmic.200400937
![]() |
[108] |
Recsei P, Kreiswirth B, O'Reilly M, et al. (1986) Regulation of exoprotein gene expression in Staphylococcus aureus by agr. Mol Gen Genet 202: 58-61. doi: 10.1007/BF00330517
![]() |
[109] | Peng HL, Novick RP, Kreiswirth B, et al. (1988) Cloning, characterization, and sequencing of an accessory gene regulator (agr) in Staphylococcus aureus. J Bacteriol 170: 4365-4372. |
[110] | Cheung AL, Projan SJ (1994) Cloning and sequencing of sarA of Staphylococcus aureus, a gene required for the expression of agr. J Bacteriol 176: 4168-4172. |
[111] | McCarthy H, Rudkin JK, Black NS, et al. (2015) Methicillin resistance and the biofilm phenotype in Staphylococcus aureus. Front Cell Infect Microbiol 5:1. |
[112] |
Liang X, Ji Y (2007) Comparative analysis of staphylococcal adhesion and internalization by epithelial cells. Meth Mol Biol (Clifton, NJ) 391: 145-151. doi: 10.1007/978-1-59745-468-1_11
![]() |
[113] |
Jin T, Bokarewa M, Foster T, et al. (2004) Staphylococcus aureus resists human defensins by production of staphylokinase, a novel bacterial evasion mechanism. J Immunol 172: 1169-1176. doi: 10.4049/jimmunol.172.2.1169
![]() |
[114] |
Lan L, Cheng A, Dunman PM, et al. (2010) Golden pigment production and virulence gene expression are affected by metabolisms in Staphylococcus aureus. J Bacteriol 192: 3068-3077. doi: 10.1128/JB.00928-09
![]() |
[115] |
Hammel M, Sfyroera G, Pyrpassopoulos S, et al. (2007) Characterization of Ehp, a secreted complement inhibitory protein from Staphylococcus aureus. J Biol Chem 282: 30051-30061. doi: 10.1074/jbc.M704247200
![]() |
[116] |
de Haas CJ, Veldkamp KE, Peschel A, et al. (2004) Chemotaxis inhibitory protein of Staphylococcus aureus, a bacterial antiinflammatory agent. J Exp Med 199: 687-695. doi: 10.1084/jem.20031636
![]() |
[117] | Rogolsky M (1979) Nonenteric toxins of Staphylococcus aureus. Microbiol Rev 43: 320-360. |
[118] | Prevost G, Mourey L, Colin DA, et al. (2001) Staphylococcal pore-forming toxins. Curr Top Microbiol Immunol 257: 53-83. |
[119] |
Herbert S, Ziebandt AK, Ohlsen K, et al. (2010) Repair of global regulators in Staphylococcus aureus 8325 and comparative analysis with other clinical isolates. Infect Immun 78: 2877-2889. doi: 10.1128/IAI.00088-10
![]() |
[120] |
Pocsfalvi G, Cacace G, Cuccurullo M, et al. (2008) Proteomic analysis of exoproteins expressed by enterotoxigenic Staphylococcus aureus strains. Proteomics 8: 2462-2476. doi: 10.1002/pmic.200700965
![]() |
[121] |
Dinges MM, Orwin PM, Schlievert PM (2000) Exotoxins of Staphylococcus aureus. Clin Microbiol Rev 13: 16-34. doi: 10.1128/CMR.13.1.16-34.2000
![]() |
[122] |
Defres S, Marwick C, Nathwani D (2009) MRSA as a cause of lung infection including airway infection, community-acquired pneumonia and hospital-acquired pneumonia. Eur Respir J 34: 1470-1476. doi: 10.1183/09031936.00122309
![]() |
[123] |
Watkins RR, David MZ, Salata RA (2012) Current concepts on the virulence mechanisms of methicillin-resistant Staphylococcus aureus. J Med Microbiol 61: 1179-1193. doi: 10.1099/jmm.0.043513-0
![]() |
[124] |
Bubeck Wardenburg J, Bae T, Otto M, et al. (2007) Poring over pores: Alpha-hemolysin and Panton-Valentine leukocidin in Staphylococcus aureus pneumonia. Nat Med 13: 1405-1406. doi: 10.1038/nm1207-1405
![]() |
[125] |
Parker D, Prince A (2012) Immunopathogenesis of Staphylococcus aureus pulmonary infection. Sem Immunopathol 34: 281-297. doi: 10.1007/s00281-011-0291-7
![]() |
[126] |
Gillet Y, Issartel B, Vanhems P, et al. (2002) Association between Staphylococcus aureus strains carrying gene for Panton-Valentine leukocidin and highly lethal necrotising pneumonia in young immunocompetent patients. Lancet 359: 753-759. doi: 10.1016/S0140-6736(02)07877-7
![]() |
[127] |
Labandeira-Rey M, Couzon F, Boisset S, et al. (2007) Staphylococcus aureus Panton-Valentine leukocidin causes necrotizing pneumonia. Science 315: 1130-1133. doi: 10.1126/science.1137165
![]() |
[128] |
Ragle BE, Bubeck Wardenburg J (2009) Anti-alpha-hemolysin monoclonal antibodies mediate protection against Staphylococcus aureus pneumonia. Infect Immun 77: 2712-2718. doi: 10.1128/IAI.00115-09
![]() |
[129] |
Ragle BE, Karginov VA, Bubeck Wardenburg J (2010) Prevention and treatment of Staphylococcus aureus pneumonia with a beta-cyclodextrin derivative. Antimicrob Agents Chemother 54: 298-304. doi: 10.1128/AAC.00973-09
![]() |
[130] |
Bubeck Wardenburg J, Schneewind O (2008) Vaccine protection against Staphylococcus aureus pneumonia. J Exp Med 205: 287-294. doi: 10.1084/jem.20072208
![]() |
[131] | Stulik L, Malafa S, Hudcova J, et al. (2014) Α-hemolysin activity of methicillin-susceptible S. aureus predicts ventilator-associated pneumonia. Am J Respir Crit Care Med 190: 1139-1148. |
[132] |
Löffler B, Hussain M, Grundmeier M, et al. (2010) Staphylococcus aureus Panton-Valentine leukocidin is a very potent cytotoxic factor for human neutrophils. PLoS Pathog 6: e1000715. doi: 10.1371/journal.ppat.1000715
![]() |
[133] |
Genestier AL, Michallet MC, Prevost G, et al. (2005) Staphylococcus aureus Panton-Valentine leukocidin directly targets mitochondria and induces Bax-independent apoptosis of human neutrophils. J Clin Invest 115: 3117-3127. doi: 10.1172/JCI22684
![]() |
[134] |
Melles DC, van Leeuwen WB, Boelens HA, et al. (2006) Panton-Valentine leukocidin genes in Staphylococcus aureus. Emerg Infect Dis 12: 1174-1175. doi: 10.3201/eid1207.050865
![]() |
[135] | da Silva MC, Zahm JM, Gras D, et al. (2004) Dynamic interaction between airway epithelial cells and Staphylococcus aureus. Am J Physiol 287: L543-L551. |
[136] | Hildebrand A, Pohl M, Bhakdi S (1991) Staphylococcus aureus alpha-toxin. Dual mechanism of binding to target cells. J Biol Chem 266: 17195-17200. |
[137] | Tweten RK, Christianson KK, Iandolo JJ (1983) Transport and processing of staphylococcal alpha-toxin. J Bacteriol 156: 524-528. |
[138] | Gray GS, Kehoe M (1984) Primary sequence of the alpha-toxin gene from Staphylococcus aureus Wood 46. Infect Immun 46: 615-618. |
[139] |
Schwiering M, Brack A, Stork R, et al. (2013) Lipid and phase specificity of alpha-toxin from S. aureus. Biochim Biophys Acta 1828: 1962-1972. doi: 10.1016/j.bbamem.2013.04.005
![]() |
[140] |
Valeva A, Hellmann N, Walev I, et al. (2006) Evidence that clustered phosphocholine head groups serve as sites for binding and assembly of an oligomeric protein pore. J Biol Chem 281: 26014-26021. doi: 10.1074/jbc.M601960200
![]() |
[141] |
Galdiero S, Gouaux E (2004) High resolution crystallographic studies of alpha-hemolysin-phospholipid complexes define heptamer-lipid head group interactions: Implication for understanding protein-lipid interactions. Prot Sci 13: 1503-1511. doi: 10.1110/ps.03561104
![]() |
[142] |
Wilke GA, Bubeck Wardenburg J (2010) Role of a disintegrin and metalloprotease 10 in Staphylococcus aureus alpha-hemolysin-mediated cellular injury. Proc Nat Acad Sci USA 107: 13473-13478. doi: 10.1073/pnas.1001815107
![]() |
[143] |
Inoshima I, Inoshima N, Wilke GA, et al. (2011) A Staphylococcus aureus pore-forming toxin subverts the activity of ADAM10 to cause lethal infection in mice. Nat Med 17: 1310-1314. doi: 10.1038/nm.2451
![]() |
[144] |
Pany S, Vijayvargia R, Krishnasastry MV (2004) Caveolin-1 binding motif of alpha-hemolysin: Its role in stability and pore formation. Biochem Biophys Res Commun 322: 29-36. doi: 10.1016/j.bbrc.2004.07.073
![]() |
[145] |
Berube BJ, Bubeck Wardenburg J (2013) Staphylococcus aureus alpha-toxin: Nearly a century of intrigue. Toxins (Basel) 5: 1140-1166. doi: 10.3390/toxins5061140
![]() |
[146] | Korchev YE, Alder GM, Bakhramov A, et al. (1995) Staphylococcus aureus alpha-toxin-induced pores: Channel-like behavior in lipid bilayers and patch clamped cells. J Membr Biol 143: 143-151. |
[147] |
Krasilnikov OV, Merzlyak PG, Yuldasheva LN, et al. (2000) Electrophysiological evidence for heptameric stoichiometry of ion channels formed by Staphylococcus aureus alpha-toxin in planar lipid bilayers. Mol Microbiol 37: 1372-1378. doi: 10.1046/j.1365-2958.2000.02080.x
![]() |
[148] |
Gouaux JE, Braha O, Hobaugh MR, et al. (1994) Subunit stoichiometry of staphylococcal alpha-hemolysin in crystals and on membranes: A heptameric transmembrane pore. Proc Nat Acad Sci USA 91: 12828-12831. doi: 10.1073/pnas.91.26.12828
![]() |
[149] |
Gouaux E (1998) Alpha-hemolysin from Staphylococcus aureus: An archetype of beta-barrel, channel-forming toxins. J Struct Biol 121: 110-122. doi: 10.1006/jsbi.1998.3959
![]() |
[150] |
Montoya M, Gouaux E (2003) Beta-barrel membrane protein folding and structure viewed through the lens of alpha-hemolysin. Biochim Biophys Acta 1609: 19-27. doi: 10.1016/S0005-2736(02)00663-6
![]() |
[151] |
Jayasinghe L, Miles G, Bayley H (2006) Role of the amino latch of staphylococcal alpha-hemolysin in pore formation: A co-operative interaction between the N terminus and position 217. J Biol Chem 281: 2195-2204. doi: 10.1074/jbc.M510841200
![]() |
[152] |
Valeva A, Walev I, Pinkernell M, et al. (1997) Transmembrane beta-barrel of staphylococcal alpha-toxin forms in sensitive but not in resistant cells. Proc Nat Acad Sci USA 94: 11607-11611. doi: 10.1073/pnas.94.21.11607
![]() |
[153] | Menzies BE, Kernodle DS (1994) Site-directed mutagenesis of the alpha-toxin gene of Staphylococcus aureus: Role of histidines in toxin activity in vitro and in a murine model. Infect Immun 62: 1843-1847. |
[154] | Bhakdi S, Tranum-Jensen J (1991) Alpha-toxin of Staphylococcus aureus. Microbiol Rev 55: 733-751. |
[155] |
Füssle R, Bhakdi S, Sziegoleit A, et al. (1981) On the mechanism of membrane damage by Staphylococcus aureus alpha-toxin. J Cell Biol 91: 83-94. doi: 10.1083/jcb.91.1.83
![]() |
[156] |
Song L, Hobaugh MR, Shustak C, et al. (1996) Structure of staphylococcal alpha-hemolysin, a heptameric transmembrane pore. Science 274: 1859-1866. doi: 10.1126/science.274.5294.1859
![]() |
[157] |
Menestrina G (1986) Ionic channels formed by Staphylococcus aureus alpha-toxin: Voltage-dependent inhibition by divalent and trivalent cations. J Membr Biol 90: 177-190. doi: 10.1007/BF01869935
![]() |
[158] |
Kasianowicz JJ, Brandin E, Branton D, et al. (1996) Characterization of individual polynucleotide molecules using a membrane channel. Proc Nat Acad Sci USA 93: 13770-13773. doi: 10.1073/pnas.93.24.13770
![]() |
[159] |
Aksimentiev A, Schulten K (2005) Imaging alpha-hemolysin with molecular dynamics: Ionic conductance, osmotic permeability, and the electrostatic potential map. Biophys J 88: 3745-3761. doi: 10.1529/biophysj.104.058727
![]() |
[160] | Walev I, Martin E, Jonas D, et al. (1993) Staphylococcal alpha-toxin kills human keratinocytes by permeabilizing the plasma membrane for monovalent ions. Infect Immun 61: 4972-4979. |
[161] |
Kloft N, Busch T, Neukirch C, et al. (2009) Pore-forming toxins activate MAPK p38 by causing loss of cellular potassium. Biochem Biophys Res Commun 385: 503-506. doi: 10.1016/j.bbrc.2009.05.121
![]() |
[162] | Jonas D, Walev I, Berger T, et al. (1994) Novel path to apoptosis: Small transmembrane pores created by staphylococcal alpha-toxin in T lymphocytes evoke internucleosomal DNA degradation. Infect Immun 62: 1304-1312. |
[163] |
Valeva A, Walev I, Gerber A, et al. (2000) Staphylococcal alpha-toxin: Repair of a calcium-impermeable pore in the target cell membrane. Mol Microbiol 36: 467-476. doi: 10.1046/j.1365-2958.2000.01865.x
![]() |
[164] |
Walev I, Palmer M, Martin E, et al. (1994) Recovery of human fibroblasts from attack by the pore-forming alpha-toxin of Staphylococcus aureus. Microb Pathogen 17: 187-201. doi: 10.1006/mpat.1994.1065
![]() |
[165] | Ahnert-Hilger G, Bhakdi S, Gratzl M (1985) Minimal requirements for exocytosis. A study using PC 12 cells permeabilized with staphylococcal alpha-toxin. J Biol Chem 260: 12730-12734. |
[166] | Suttorp N, Seeger W, Dewein E, et al. (1985) Staphylococcal alpha-toxin-induced PGI2 production in endothelial cells: Role of calcium. Am J Physiol 248: C127-C134. |
[167] |
Eichstaedt S, Gäbler K, Below S, et al. (2009) Effects of Staphylococcus aureus-hemolysin a on calcium signalling in immortalized human airway epithelial cells. Cell Calcium 45: 165-176. doi: 10.1016/j.ceca.2008.09.001
![]() |
[168] |
Gierok P, Harms M, Richter E, et al. (2014) Staphylococcus aureus alpha-toxin mediates general and cell type-specific changes in metabolite concentrations of immortalized human airway epithelial cells. PLoS ONE 9: e94818. doi: 10.1371/journal.pone.0094818
![]() |
[169] | Husmann M, Dersch K, Bobkiewicz W, et al. (2006) Differential role of p38 mitogen activated protein kinase for cellular recovery from attack by pore-forming S. aureus alpha-toxin or streptolysin O. Biochem Biophys Res Commun 344: 1128-1134. |
[170] | Ostedgaard LS, Shasby DM, Welsh MJ (1992) Staphylococcus aureus alpha-toxin permeabilizes the basolateral membrane of a Cl--secreting epithelium. Am J Physiol 263: L104-L112. |
[171] |
Dragneva Y, Anuradha CD, Valeva A, et al. (2001) Subcytocidal attack by staphylococcal alpha-toxin activates NFkappaB and induces interleukin-8 production. Infect Immun 69: 2630-2635. doi: 10.1128/IAI.69.4.2630-2635.2001
![]() |
[172] |
Lizak M, Yarovinsky TO (2012) Phospholipid scramblase 1 mediates type I interferon-induced protection against staphylococcal alpha-toxin. Cell Host Microbe 11: 70-80. doi: 10.1016/j.chom.2011.12.004
![]() |
[173] | la Sala A, Ferrari D, Di Virgilio F, et al. (2003) Alerting and tuning the immune response by extracellular nucle |
[174] |
Okada SF, Nicholas RA, Kreda SM, et al. (2006) Physiological regulation of ATP release at the apical surface of human airway epithelia. J Biol Chem 281: 22992-23002. doi: 10.1074/jbc.M603019200
![]() |
[175] | Bond S, Naus CC (2014) The pannexins: Past and present. Front Physiol 5: 1-24. |
[176] |
Russo MJ, Bayley H, Toner M (1997) Reversible permeabilization of plasma membranes with an engineered switchable pore. Nat Biotechnol 15: 278-282. doi: 10.1038/nbt0397-278
![]() |
[177] | Cassidy PS, Harshman S (1973) The binding of staphylococcal 125I-alpha-toxin (B) to erythrocytes. J Biol Chem 248: 5545-5546. |
[178] |
Maurer K, Reyes-Robles T, Alonzo F, 3rd, et al. (2015) Autophagy mediates tolerance to Staphylococcus aureus alpha-toxin. Cell Host Microbe 17: 429-440. doi: 10.1016/j.chom.2015.03.001
![]() |
[179] |
Kloft N, Neukirch C, Bobkiewicz W, et al. (2010) Pro-autophagic signal induction by bacterial pore-forming toxins. Med Microbiol Immunol 199: 299-309. doi: 10.1007/s00430-010-0163-0
![]() |
[180] |
Husmann M, Beckmann E, Boller K, et al. (2009) Elimination of a bacterial pore-forming toxin by sequential endocytosis and exocytosis. FEBS Lett 583: 337-344. doi: 10.1016/j.febslet.2008.12.028
![]() |
[181] |
Kwak YK, Vikstrom E, Magnusson KE, et al. (2012) The Staphylococcus aureus alpha-toxin perturbs the barrier function in Caco-2 epithelial cell monolayers by altering junctional integrity. Infect Immun 80: 1670-1680. doi: 10.1128/IAI.00001-12
![]() |
[182] |
Boucher RC (2004) New concepts of the pathogenesis of cystic fibrosis lung disease. Eur Respir J 23: 146-158. doi: 10.1183/09031936.03.00057003
![]() |
[183] |
Kunzelmann K, McMorran B (2004) First encounter: How pathogens compromise epithelial transport. Physiology (Bethesda, Md) 19: 240-244. doi: 10.1152/physiol.00015.2004
![]() |
[184] |
Tarran R (2004) Regulation of airway surface liquid volume and mucus transport by active ion transport. Proc Am Thoracic Soc 1: 42-46. doi: 10.1513/pats.2306014
![]() |
[185] |
Escotte S, Al Alam D, Le Naour R, et al. (2006) T cell chemotaxis and chemokine release after Staphylococcus aureus interaction with polarized airway epithelium. Am J Respir Cell Mol Biol 34: 348-354. doi: 10.1165/rcmb.2005-0191OC
![]() |
[186] |
Lee RJ, Foskett JK (2014) Ca2+ signaling and fluid secretion by secretory cells of the airway epithelium. Cell Calcium 55: 325-336. doi: 10.1016/j.ceca.2014.02.001
![]() |
[187] | Suttorp N, Hessz T, Seeger W, et al. (1988) Bacterial exotoxins and endothelial permeability for water and albumin in vitro. Am J Physiol Cell Physiol 255: C368-C376. |
[188] |
Hocke AC, Temmesfeld-Wollbrueck B, Schmeck B, et al. (2006) Perturbation of endothelial junction proteins by Staphylococcus aureus alpha-toxin: Inhibition of endothelial gap formation by adrenomedullin. Histochem Cell Biol 126: 305-316. doi: 10.1007/s00418-006-0174-5
![]() |
[189] |
Stull JT, Tansey MG, Tang DC, et al. (1993) Phosphorylation of myosin light chain kinase: A cellular mechanism for Ca2+ desensitization. Mol Cell Biochem 127-128: 229-237. doi: 10.1007/BF01076774
![]() |
[190] | Horiuchi K, Le Gall S, Schulte M, et al. (2007) Substrate selectivity of epidermal growth factor-receptor ligand sheddases and their regulation by phorbol esters and calcium influx. Mol Biol Cell 18: 176-188. |
[191] |
Le Gall SM, Bobe P, Reiss K, et al. (2009) ADAMs 10 and 17 represent differentially regulated components of a general shedding machinery for membrane proteins such as transforming growth factor alpha, L-selectin, and tumor necrosis factor alpha. Mol Biol Cell 20: 1785-1794. doi: 10.1091/mbc.E08-11-1135
![]() |
[192] |
Brieher WM, Yap AS (2013) Cadherin junctions and their cytoskeleton(s). Curr Opin Cell Biol 25: 39-46. doi: 10.1016/j.ceb.2012.10.010
![]() |
[193] |
Zaidel-Bar R, Itzkovitz S, Ma'ayan A, et al. (2007) Functional atlas of the integrin adhesome. Nat Cell Biol 9: 858-867. doi: 10.1038/ncb0807-858
![]() |
[194] | Dreymueller D, Uhlig S, Ludwig A (2015) ADAM-family metalloproteinases in lung inflammation: Potential therapeutic targets. Am J Physiol 308: L325-L343. |
[195] |
Sahin U, Weskamp G, Kelly K, et al. (2004) Distinct roles for ADAM10 and ADAM17 in ectodomain shedding of six EGFR ligands. J Cell Biol 164: 769-779. doi: 10.1083/jcb.200307137
![]() |
[196] |
Park PW, Foster TJ, Nishi E, et al. (2004) Activation of syndecan-1 ectodomain shedding by Staphylococcus aureus alpha-toxin and beta-toxin. J Biol Chem 279: 251-258. doi: 10.1074/jbc.M308537200
![]() |
[197] |
Hayashida A, Bartlett AH, Foster TJ, et al. (2009) Staphylococcus aureus beta-toxin induces lung injury through syndecan-1. Am J Pathol 174: 509-518. doi: 10.2353/ajpath.2009.080394
![]() |
[198] |
Tengholm A, Hellman B, Gylfe E (2000) Mobilization of Ca2+ stores in individual pancreatic beta-cells permeabilized or not with digitonin or alpha-toxin. Cell Calcium 27: 43-51. doi: 10.1054/ceca.1999.0087
![]() |
[199] |
Huang TY, Minamide LS, Bamburg JR, et al. (2008) Chronophin mediates an ATP-sensing mechanism for cofilin dephosphorylation and neuronal cofilin-actin rod formation. Dev Cell 15: 691-703. doi: 10.1016/j.devcel.2008.09.017
![]() |
[200] |
Eichstaedt S, Gäbler K, Below S, et al. (2008) Phospholipase C-activating plasma membrane receptors and calcium signaling in immortalized human airway epithelial cells. J Recept Signal Transd 28: 591-612. doi: 10.1080/10799890802407120
![]() |
[201] |
Schwiebert EM, Zsembery A (2003) Extracellular ATP as a signaling molecule for epithelial cells. Biochim Biophys Acta 1615: 7-32. doi: 10.1016/S0005-2736(03)00210-4
![]() |
[202] |
Tarran R, Button B, Boucher RC (2006) Regulation of normal and cystic fibrosis airway surface liquid volume by phasic shear stress. Annu Rev Physiol 68: 543-561. doi: 10.1146/annurev.physiol.68.072304.112754
![]() |
[203] |
Evans JH, Sanderson MJ (1999) Intracellular calcium oscillations regulate ciliary beat frequency of airway epithelial cells. Cell Calcium 26: 103-110. doi: 10.1054/ceca.1999.0060
![]() |
[204] |
Yun YS, Min YG, Rhee CS, et al. (1999) Effects of alpha-toxin of Staphylococcus aureus on the ciliary activity and ultrastructure of human nasal ciliated epithelial cells. Laryngoscope 109: 2021-2024. doi: 10.1097/00005537-199912000-00024
![]() |
[205] |
Knowles M, Robinson J, Wood R, et al. (1997) Ion composition of airway surface liquid of patients with cystic fibrosis as compared with normal and disease-control subjects. J Clin Invest 100: 2588-2595. doi: 10.1172/JCI119802
![]() |
[206] |
Olson R, Nariya H, Yokota K, et al. (1999) Crystal structure of staphylococcal LukF delineates conformational changes accompanying formation of a transmembrane channel. Nat Struct Biol 6: 134-140. doi: 10.1038/5821
![]() |
° | a | b | c | d |
a | {a} | {a} | {a} | {a} |
b | {a} | {a} | {a} | {a} |
c | {a} | {a} | {a} | {a, b} |
d | {a} | {a} | {a, b} | {a, b, c} |
![]() |
$ a $ | $ b $ | $ c $ | $ d $ |
$ \mu_{\mathcal{A}} $ | $ 0.9 $ | $ 0.6 $ | $ 0.8 $ | $ 0.5 $ |
$ \lambda_{\mathcal{A}} $ | $ 0.5 $ | $ 0.8 $ | $ 0.7 $ | $ 0.9 $ |