Loading [MathJax]/jax/element/mml/optable/MathOperators.js
Review

Bone remodeling and biological effects of mechanical stimulus

  • Received: 05 November 2019 Accepted: 17 January 2020 Published: 19 January 2020
  • This review describes the physiology of normal bone tissue as an introduction to the subsequent discussion on bone remodeling and biomechanical stimulus. As a complex architecture with heterogeneous and anisotropic hierarchy, the skeletal bone has been anatomically analysed with different levelling principles, extending from nano- to the whole bone scale. With the interpretation of basic bone histomorphology, the main compositions in bone are summarized, including various organic proteins in the bone matrix and inorganic minerals as the reinforcement. The cell populations that actively participate in the bone remodeling—osteoclasts, osteoblasts and osteocytes—have also been discussed since they are the main operators in bone resorption and formation. A variety of factors affect the bone remodeling, such as hormones, cytokines, mechanical stimulus and electromagnetic stimulus. As a particularly potent stimulus for bone cells, mechanical forces play a crucial role in enhancing bone strength and preventing bone loss with aging. By combing all these aspects together, the information lays the groundwork for systematically understanding the link between bone physiology and orchestrated process of mechanically mediated bone homoestasis.

    Citation: Chao Hu, Qing-Hua Qin. Bone remodeling and biological effects of mechanical stimulus[J]. AIMS Bioengineering, 2020, 7(1): 12-28. doi: 10.3934/bioeng.2020002

    Related Papers:

    [1] Hongyan Wang, Shaoping Jiang, Yudie Hu, Supaporn Lonapalawong . Analysis of drug-resistant tuberculosis in a two-patch environment using Caputo fractional-order modeling. AIMS Mathematics, 2024, 9(11): 32696-32733. doi: 10.3934/math.20241565
    [2] Saima Rashid, Yolanda Guerrero Sánchez, Jagdev Singh, Khadijah M Abualnaja . Novel analysis of nonlinear dynamics of a fractional model for tuberculosis disease via the generalized Caputo fractional derivative operator (case study of Nigeria). AIMS Mathematics, 2022, 7(6): 10096-10121. doi: 10.3934/math.2022562
    [3] Hasib Khan, Jehad Alzabut, J.F. Gómez-Aguilar, Praveen Agarwal . Piecewise mABC fractional derivative with an application. AIMS Mathematics, 2023, 8(10): 24345-24366. doi: 10.3934/math.20231241
    [4] Cicik Alfiniyah, Wanwha Sonia Putri Artha Soetjianto, Ahmadin, Muhamad Hifzhudin Noor Aziz, Siti Maisharah Sheikh Ghadzi . Mathematical modeling and optimal control of tuberculosis spread among smokers with case detection. AIMS Mathematics, 2024, 9(11): 30472-30492. doi: 10.3934/math.20241471
    [5] Qun Dai, Longkun Zhang . A dual delays epidemic model for TB with adaptive mobility behavior. AIMS Mathematics, 2025, 10(7): 15231-15263. doi: 10.3934/math.2025683
    [6] Manvendra Narayan Mishra, Faten Aldosari . Comparative study of tuberculosis infection by using general fractional derivative. AIMS Mathematics, 2025, 10(1): 1224-1247. doi: 10.3934/math.2025058
    [7] Kamel Guedri, Rahat Zarin, and Mowffaq Oreijah . Evaluating the impact of vaccination and progression delays on tuberculosis dynamics with disability outcomes: A case study in Saudi Arabia. AIMS Mathematics, 2025, 10(4): 7970-8001. doi: 10.3934/math.2025366
    [8] Alessandra Jannelli, Maria Paola Speciale . On the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations. AIMS Mathematics, 2021, 6(8): 9109-9125. doi: 10.3934/math.2021529
    [9] J. Vanterler da C. Sousa, E. Capelas de Oliveira, L. A. Magna . Fractional calculus and the ESR test. AIMS Mathematics, 2017, 2(4): 692-705. doi: 10.3934/Math.2017.4.692
    [10] Fawaz K. Alalhareth, Seham M. Al-Mekhlafi, Ahmed Boudaoui, Noura Laksaci, Mohammed H. Alharbi . Numerical treatment for a novel crossover mathematical model of the COVID-19 epidemic. AIMS Mathematics, 2024, 9(3): 5376-5393. doi: 10.3934/math.2024259
  • This review describes the physiology of normal bone tissue as an introduction to the subsequent discussion on bone remodeling and biomechanical stimulus. As a complex architecture with heterogeneous and anisotropic hierarchy, the skeletal bone has been anatomically analysed with different levelling principles, extending from nano- to the whole bone scale. With the interpretation of basic bone histomorphology, the main compositions in bone are summarized, including various organic proteins in the bone matrix and inorganic minerals as the reinforcement. The cell populations that actively participate in the bone remodeling—osteoclasts, osteoblasts and osteocytes—have also been discussed since they are the main operators in bone resorption and formation. A variety of factors affect the bone remodeling, such as hormones, cytokines, mechanical stimulus and electromagnetic stimulus. As a particularly potent stimulus for bone cells, mechanical forces play a crucial role in enhancing bone strength and preventing bone loss with aging. By combing all these aspects together, the information lays the groundwork for systematically understanding the link between bone physiology and orchestrated process of mechanically mediated bone homoestasis.


    The paper [1] by Pawlak was the first article focused on the rough area between the interior set A and the closure set ¯A of a subset A in a universal set X. This idea led to many applications in decision theory. The theory of rough sets is constructed using the equivalence classes as its building blocks.

    The most efficacious tools to study the generalization of rough set theory are the neighborhood systems. The main idea in this theory is the upper and lower approximations that have been defined using different types of neighborhoods instead of equivalence classes such as left and right neighborhoods [2,3,4,5], minimal left neighborhoods [6] and minimal right neighborhoods [7], and the intersection of minimal left and right neighborhoods [8]. Afterwards, the approximations by minimal right neighborhoods which are determined by reflexive relations that form the base of the topological space defined in [9]. In 2018, Dai et al. [10] presented new kind of neighborhoods, namely the maximal right neighborhoods which were determined by similarity relations and have been used to propose three new kinds of approximations. Dai et al.'s approximations [10] differed from Abo-Tabl's approximations [9] in that the corresponding upper and lower approximations, boundary regions, accuracy measures, and roughness measures in two types of Dai et al.'s approximations [10] had a monotonicity. Later on, Al-shami [11] embraced a new type of neighborhood systems namely, the intersection of maximal right and left neighborhoods, and then used this type to present new approximations. These approximations improved the accuracy measures more than Dai et al.'s approximations [10]. Al-shami's [11] accuracy measures preserved the monotonic property under any arbitrary relation. The paper [12], by Molodtsov, was the first article that defined the notion "soft set", and it has many applications in uncertainty area or ambiguity decision. A theoretical research on soft set theory was given in [13] by Maji et al. The paper [14] by Ali et al. proposed many soft set-theoretical notions such as union, intersection, difference and complement. [15,16,17,18,19,20] objected to developing the theory and the applications of soft sets. In [21], the authors introduced the soft ideal notion. It is a completely new approach for modeling vagueness and uncertainty by reducing the boundary region and increasing the accuracy of a rough set which helped scholars to solve many real-life problems [4,22,23,24,25]. Recently, many extensions of the classical rough set approximations have been applied to provide new rough paradigms using certain topological structures and concepts like subset neighborhoods, containment neighborhoods, and maximal and minimal neighborhoods to deal with rough set notions and address some real-life problems [2,4,26,27,28]. Numerous researchers have recently examined some topological concepts, including continuity, separation axioms, closure spaces, and connectedness in ideal approximation spaces [29,30,31]. Ordinary rough sets were defined using an equivalence relation R on X, and produced two approximations, one is lower and second is upper. The space (X,R) is named approximation space. In the soft case, soft roughness used soft relations [32]. Some researchers transferred the common definitions in set-topology to soft set-topology, depending on that soft topology is an extension to the usual topology as explained in [15]. Many researchers objected to the basics of set-topology and subsequently the well-known embedding theorems but in point of view of soft set-topology with some real-life applications (see [33,34,35,36,37,38]). This paper used the notion of soft binary relations to ensure that the soft interior and soft closure in approximation spaces utilizing soft ideal to generate soft ideal approximation topological spaces based on soft minimal neighborhoods. We illustrated that soft rough approximations [17] are special cases of the current soft ideal approximations. Soft accumulation points, soft exterior sets, soft dense sets, and soft nowhere dense sets with respect to these spaces were defined and studied, and we gave some examples. We introduce and study soft ideal accumulation points in such spaces under a soft ideal defined on the given soft ideal. Soft separation axioms with respect to these soft ideal approximation spaces are reformulated via soft relational concepts and compared with examples to show their implications. In addition, we reformulate and study soft connectedness in these soft ideal approximation spaces. Finally, we defined soft boundary region and soft accuracy measure with respect to our soft ideal approximation spaces. We added two real life examples to illustrate the importance of the results obtained in this paper.

    This paper is divided into 6 sections beyond the introduction and the preliminaries. Section 3 defined the soft approximation spaces using a soft ideal. Section 4 is the main section of the manuscript and displays the properties of soft sets in the soft ideal approximation spaces. It has been generated using the concepts of R<x>R, soft neighborhoods and soft ideals. We study the main properties in soft ideal approximation spaces which are generalizations of the same properties of ideal approximation spaces given by Abbas et al. [31] and provide various illustrative examples. Section 5 introduced soft lower separation axioms via soft binary relations and soft ideal as a generalization of lower separation axioms given in [31]. We scrutinized its essential characterizations of some of its relationships associated with the soft ideal closure operators. Some illustrative examples are given. Section 6 reformulated and studied soft connectedness in [31] with respect to these soft ideal approximation spaces. Some examples are submitted to explain the definitions. Section 7 is devoted to comparing between the current purposed methods in Definitions 3.4–3.6 and to demonstrate that the method given in Definition 3.6 is the best in terms of developing the soft approximation operators and the values of soft accuracy. That is, the third approach in Definition 3.6 produces soft accuracy measures of soft subsets higher than their counterparts displayed in previous method 2.4 in [17]. Moreover, we applied these approaches to handle real-life problems. Section 8 is the conclusion.

    Through this paper, X stands for the universal set of objects, E denotes the set of parameters, LE denotes for a soft ideal, RE as a soft binary relation, P(X) represents all subsets of X, and SS(X) refers to the set of all soft subsets of X. All basic notions and notations of soft sets are found in [12,13,15,39,40].

    If (F,E) is a soft set of X and xX, then xˇ(F,E) whenever xF(e) for each eE. A soft set (F,E) of X with F(e)={x} for each eE is called a singleton soft set or a soft point and it is represented by xE or (x,E). Let (F1,E),(F2,E)ˇSS(X)E. Then, (F1,E) is a soft subset of (F2,E), represented by (F1,E)(F2,E), if F1(e)F2(e),eE. In that case, (F1,E) is called a soft subset of (F2,E) and (F2,E) is said to be a soft supset of (F1,E), (F2,E)(F1,E). Two soft subset (F1,E) and (F2,E) over X are called equal if (F1,E) is soft subset of (F2,E) and (F1,E) is soft supset of (F2,E). A soft set (F,E) over X is called a NULL soft set written as Φ if for each eE,F(e)=ϕ. Let A be a non-empty subset of X, then ˜AE or ˜A represents the absolute soft set (A,E) of X in which A(e)=A, for each eE. The soft intersection (resp. soft union) of (F1,E) and (F2,E) over X denoted by (F1F2,E) (resp. (F1F2,E)) and defined as (F1F2)(e)=F1(e)F2(e) (resp. (F1F2)(e)=F1(e)F2(e)) for each eE. Complementing a soft set (F,E) is represented by (F,E)c and it is defined as (F,E)c=(Fc,E) where Fc:EP(X) is a mapping defined by Fc(e)=XF(e) for all eE, and Fc is then a soft complement function of F.

    Definition 2.1. [32] Let (R,E)=RE be a soft set of X×X, that is R:EP(X×X). Then, RE is said to be a soft binary relation of X. RE is a collection of parameterized binary relations of X, from that R(e) is a binary relation on X for all parameters eE. The set of all soft binary relations of X is denoted by SBr(X).

    Definition 2.2. [15] Let ˜τ be a collection of soft sets over a universe X with a fixed set of parameters E. Then, ˜τSS(X)E is called a soft topology on X if

    (1)˜X,ΦEˇ˜τ,

    (2) the intersection of any two soft sets in ˜τ belongs to ˜τ,

    (2) the union of any number of soft sets in in ˜τ belongs to ˜τ.

    The triplet (X,˜τ,E) is called a soft topological space over X.

    Definition 2.3. [28] A mapping Cl:SS(X)ESS(X)E is called a soft closure operator on X if it satisfies these properties for every (F,E),(G,E)ˇSS(X)E:

    (1) Cl(Φ)=Φ,

    (2) (F,E)Cl(F,E),

    (3) Cl[(F,E)(G,E)]=Cl(F,E)Cl(G,E),

    (4) Cl(Cl(F,E))=Cl(F,E).

    Definition 2.4. [17] Let R:EP(X1×X2) and AX2. Then, the sets R_A(e),¯RA(e) could be defined by

    R_A(e)={xX1:ϕxR(e)A},¯RA(e)={xX1:xR(e)Aϕ}

    where xR(e)={yX2:(x,y)R(e)}. Moreover, R_:EP(X1) and ¯R:EP(X1) and we say (X1,X2,R) a generalized soft approximation space.

    Definition 2.5. [21] Let LE be a non-empty family of soft sets of X. Then, LESS(X)E is said to be a soft ideal on X if the following properties are fulfilled:

    (1)ΦˇLE,

    (2) (F,E)ˇLE and (G,E)(F,E) imply (G,E)ˇLE,

    (3) (F,E),(G,E)ˇLE imply (F,E)(G,E)ˇLE.

    In this section, we define the soft approximation spaces using soft ideals.

    Definition 3.1. Let RE be a soft binary relation of X and (x,y)X×X. Then, (x,y)ˇR whenever (x,y)R(e) for each eE.

    Definition 3.2. Let RE be a soft binary relation of X. Then, the soft afterset of xˇ˜X is xR={yˇ˜X:(x,y)ˇR}. Also, the soft foreset of xˇ˜X is Rx={yˇ˜X:(y,x)ˇR}.

    Definition 3.3. Let RE be a soft binary relation over X. Then, a soft set <x>R:EP(X) is defined by

    <x>R={xyR(yR)ify:xˇyR,Φo.w.

    Also, R<x>: EP(X) is the intersection of all foresets containing x, that is,

    R<x>={xyR(Ry)ify:xˇRy,Φo.w.

    Also, R<x>R=R<x><x>R.

    Lemma 3.1. Let RE be a soft binary relation over X. Then,

    (1) If xˇ<y>R, then <x>R⊑<y>R.

    (2) If xˇR<y>R, then R<x>RR<y>R.

    Proof. (1) Let zˇ<x>R=xˇwR(wR). Then, z is contained in any wR which contain x, and since x is contained in any uR which contains y, we have zˇ<y>R. Hence, <x>R⊑<y>R.

    (2) Straightforward from part (1).

    Definition 3.4. Let RE be a soft binary relation of X. For a soft set (F,E)ˇSS(X)E, the soft lower approximation Apr_1S(F,E) and the soft upper approximation ¯Apr1S(F,E) are defined by:

    Apr_1S(F,E)={xˇ(F,E): <x>R(F,E)}, (3.1)
    ¯Apr1S(F,E)=(F,E){xˇ˜X: <x>R(F,E)Φ}. (3.2)

    Theorem 3.1. Let (F,E),(G,E)ˇSS(X)E. The soft upper approximation defined by Eq (3.2) has the following properties:

    (1) ¯Apr1S(Φ)=Φ and ¯Apr1S(˜X)=˜X,

    (2) (F,E)¯Apr1S(F,E),

    (3) (F,E)(G,E)¯Apr1S(F,E)¯Apr1S(G,E),

    (4) ¯Apr1S[(F,E)(G,E)]¯Apr1S(F,E)¯Apr1S(G,E),

    (5) ¯Apr1S[(F,E)(G,E)]=¯Apr1S(F,E)¯Apr1S(G,E),

    (6) ¯Apr1S(¯Apr1S(F,E))=¯Apr1S(F,E),

    (7) ¯Apr1S(F,E)=[Apr_1S(F,E)c]c.

    Proof. (1),(2) It is clear from Definition 3.4.

    (3) Let xˇ¯Apr1S[(F,E). Then, <x>R(F,E)Φ. Since (F,E)(G,E), <x>R(G,E)Φ. Therefore, xˇ¯Apr1S(G,E). Hence, ¯Apr1S(F,E)¯Apr1S(G,E).

    (4) Immediately by part (3).

    (5) ¯Apr1S[(F,E)(G,E)]=[(F,E)(G,E)]{xˇ˜X: <x>R[(F,E)(G,E)]Φ}. Then,

    ¯Apr1S[(F,E)(G,E)]=[(F,E){xˇ˜X: <x>R(F,E)Φ}][(G,E){xˇ˜X: <x>R(G,E)Φ}]. Hence, ¯Apr1S[(F,E)(G,E)]=¯Apr1S((F,E))¯Apr1S((G,E)).

    (6) From part (2), we have ¯Apr1S(F,E)¯Apr1S(¯Apr1S(F,E)).

    Conversely, let xˇ¯Apr1S(¯Apr1S(F,E)). Then, <x>R¯Apr1S(F,E)Φ. Thus, there exists yˇ<x>R¯Apr1S(F,E). That means <y>R⊑<x>R (by Lemma 3.1 part (1)) and <x>R(F,E)Φ. Hence, xˇ¯Apr1S(F,E). This completes the proof.

    (7)

    [Apr_1S(F,E)c]c=[(F,E)c{xˇ˜X:<x>R(F,E)c}]c=(F,E){xˇ˜X:<x>R(F,E)Φ}=¯Apr1S(F,E).

    Example 3.1. Let X={a,b,c,d}, E={e1,e2} and

    RE={(e1,{(a,a),(a,b),(b,d),(c,d),(d,c),(d,d)),(e2,{(a,a),(a,b),(a,c),(b,d),(b,c),(c,d),(d,c),(d,d),(d,b))}}. Then, we have

    <a>R=<b>R={(e1,{a,b}),(e2,{a,b})},<c>R={(e1,{c,d}),(e2,{c,d})},

    <d>R={(e1,{d}),(e2,{d})}. Suppose (F1,E)={(e1,{a,c}),(e2,{a,c})} and

    (F2,E)={(e1,{a,d}),(e2,{a,d})}. Therefore,

    ¯Apr1S(F1,E)=(F,E){xˇ˜X:<x>R(F,E)Φ}={(e1,{a,b,c}),(e2,{a,b,c})}, ¯Apr1S(F2,E)=˜X and ¯Apr1S[(F1,E)(F2,E)]={(e1,{a,b}),(e2,{a,b})}. Hence, ¯Apr1S[(F1,E)(F2,E)]¯Apr1S(F1,E)¯Apr1S(F2,E).

    Corollary 3.1. Let RE be a soft binary relation of X. Then, the soft operator ¯Apr1S:SS(X)ESS(X)E is said to be a soft closure operator and (X,¯Apr1S) is standing for a soft closure space. Moreover, it induces a soft topology on X written as ˜τ1S and defined by ˜τ1S={(F,E)ˇSS(X)E:¯Apr1S(F,E)c=(F,E)c}.

    Theorem 3.2. Let (F,E),(G,E)ˇSS(X)E. The soft lower approximation defined by Eq (3.1) has the following properties:

    (1) Apr_1S(Φ)=Φ and Apr_1S(˜X)=˜X,

    (2) Apr_1S(F,E)(F,E),

    (3) (F,E)(G,E)Apr_1S(F,E)Apr_1S(G,E),

    (4) Apr_1S[(F,E)(G,E)]=Apr_1S(F,E)Apr_1S(G,E),

    (5) Apr_1S[(F,E)(G,E)]Apr_1S(F,E)Apr_1S(G,E),

    (6) Apr_1S(Apr_1S(F,E))=Apr_1S(F,E),

    (7) Apr_1S(F,E)=[¯Apr1S(F,E)c]c.

    Proof. It is the same as given in Theorem 3.1.

    Note that the equality in Theorem 3.2 part (5) did not hold in general (see Example 3.1).

    Take (F1,E)={(e1,{b,c}),(e2,{b,c})} and (F2,E)={(e1,{b,d}),(e2,{b,d})}. Then,

    Apr_1S(F1,E)={xˇ(F1,E):<x>R(F1,E)}=Φ, Apr_1S(F2,E)={(e1,{d}),(e2,{d})} and

    Apr_1S[(F1,E)(F2,E)]={(e1,{c,d}),(e2,{c,d})}, which means that

    Apr_1S[(F,E)(G,E)]Apr_1S(F,E)Apr_1S(G,E).

    Definition 3.5. Let RE be a soft binary relation over X and LE a soft ideal on X. For any soft set (F,E)ˇSS(X)E, the soft lower approximation and the soft upper approximation of (F,E) by LE, denoted by Apr_2S(F,E) and ¯Apr2S(F,E) are defined by:

    Apr_2S(F,E)={xˇ(F,E):<x>R(F,E)cˇLE}, (3.3)
    ¯Apr2S(F,E)=(F,E){xˇ˜X:<x>R(F,E)ˇLE}. (3.4)

    Theorem 3.3. Let (F,E),(G,E)ˇSS(X)E. The soft upper approximation defined by Eq (3.4) has the following properties:

    (1) ¯Apr2S(Φ)=Φ and ¯Apr2S(˜X)=˜X,

    (2) (F,E)¯Apr2S(F,E),

    (3) (F,E)(G,E)¯Apr2S(F,E)¯Apr2S(G,E),

    (4) ¯Apr2S[(F,E)(G,E)]¯Apr2S(F,E)¯Apr2S(G,E),

    (5) ¯Apr2S[(F,E)(G,E)]=¯Apr2S(F,E)¯Apr2S(G,E),

    (6) ¯Apr2S(¯Apr2S(F,E))=¯Apr2S(F,E),

    (7) ¯Apr2S(F,E)=[Apr_2S(F,E)c]c.

    Proof. (1),(2) Direct from Definition 3.5.

    (3) Let xˇ¯Apr2S[(F,E). Thus, <x>R(F,E)ˇLE. Since (F,E)(G,E) and LE is a soft ideal, <x>R(G,E)ˇLE. Therefore, xˇ¯Apr2S(G,E). Hence, ¯Apr2S(F,E)¯Apr2S(G,E).

    (4) Straightforward by part (3).

    (5) ¯Apr2S[(F,E)(G,E)]=[(F,E)(G,E)]{xˇ˜X:<x>R[(F,E)(G,E)]ˇLE}. Then, ¯Apr2S[(F,E)(G,E)]=[(F,E){xˇ˜X:<x>R(F,E)ˇLE}][(G,E){xˇ˜X:<x>R(G,E)ˇLE}]. Hence, ¯Apr2S[(F,E)(G,E)]=¯Apr2S((F,E))¯Apr2S((G,E)).

    (6) From part (2), we have ¯Apr2S(F,E)¯Apr2S(¯Apr2S(F,E)).

    Conversely, let xˇ¯Apr2S(¯Apr2S(F,E)). Then, <x>R¯Apr2S(F,E)ˇLE. Therefore, <x>R¯Apr1S(F,E)Φ. Thus, there exists yˇ<x>R¯Apr2S(F,E). That means <y>R⊑<x>R (by Lemma 3.1 part (1)) and <y>R(F,E)ˇLE. Then, <x>R(F,E)ˇLE. Hence, xˇ¯Apr2S(F,E). This completes the proof.

    (7)

    [Apr_2S(F,E)c]c=[(F,E)c{xˇ˜X:<x>R(F,E)ˇLE}]c=(F,E){xˇ˜X:<x>R(F,E)ˇLE}=¯Apr2S(F,E).

    Corollary 3.2. Let RE be a soft binary relation over X and LE be a soft ideal on X. Then, the soft operator ¯Apr2S:SS(X)ESS(X)E is said to be a soft closure operator and (X,¯Apr2S) is standing for a soft closure space. Moreover, it induces a soft topology on X written as ˜τ2S and defined by ˜τ2S={(F,E)ˇSS(X)E:¯Apr2S(F,E)c=(F,E)c}.

    Theorem 3.4. Let (F,E),(G,E)ˇSS(X)E. The soft lower approximation defined by Eq (3.3) has the following properties:

    (1) Apr_2S(Φ)=Φ and Apr_2S(˜X)=˜X,

    (2) Apr_2S(F,E)(F,E),

    (3) (F,E)(G,E)Apr_2S(F,E)Apr_2S(G,E),

    (4) Apr_2S[(F,E)(G,E)]=Apr_2S(F,E)Apr_2S(G,E),

    (5) Apr_2S[(F,E)(G,E)]Apr_2S(F,E)Apr_2S(G,E),

    (6) Apr_2S(Apr_2S(F,E))=Apr_2S(F,E),

    (7) Apr_2S(F,E)=[¯Apr2S(F,E)c]c.

    Proof. It is similar to that was given in Theorem 3.3.

    Definition 3.6. Let RE be a soft binary relation over X and LE be a soft ideal on X. For any soft set (F,E)ˇSS(X)E, the soft lower approximation and soft upper approximation of (F,E) by LE, denoted by Apr_3S(F,E) and ¯Apr3S(F,E) are defined by:

    Apr_3S(F,E)={xˇ(F,E):R<x>R(F,E)cˇLE}, (3.5)
    ¯Apr3S(F,E)=(F,E){xˇ˜X:R<x>R(F,E)ˇLE}. (3.6)

    Theorem 3.5. Let (F,E),(G,E)ˇSS(X)E. The soft upper approximation defined by Eq (3.6) has the following properties:

    (1) ¯Apr3S(Φ)=Φ and ¯Apr3S(˜X)=˜X,

    (2) (F,E)¯Apr3S(F,E),

    (3) (F,E)(G,E)¯Apr3S(F,E)¯Apr3S(G,E),

    (4) ¯Apr3S[(F,E)(G,E)]¯Apr3S(F,E)¯Apr3S(G,E),

    (5) ¯Apr3S[(F,E)(G,E)]=¯Apr3S(F,E)¯Apr3S(G,E),

    (6) ¯Apr3S(¯Apr3S(F,E))=¯Apr3S(F,E),

    (7) ¯Apr3S(F,E)=[Apr_3S(F,E)c]c.

    Proof. It is clear from Theorem 3.3.

    Corollary 3.3. Let RE be a soft binary relation over X and LE be a soft ideal on X. Then, the soft operator ¯Apr3S:SS(X)ESS(X)E is said to be a soft closure operator and (X,¯Apr3S) is standing for a soft closure space. In addition, (X,RE,LE) is said to be a soft ideal approximation space. Moreover, it induces a soft topology on X written as ˜τ3S and defined by ˜τ3S={(F,E)ˇSS(X)E:¯Apr3S(F,E)c=(F,E)c}. It is clear that ˜τ1S˜τ2S˜τ3S.

    Theorem 3.6. Let (F,E),(G,E)ˇSS(X)E. The soft lower approximation defined by Eq (3.5) has the following properties:

    (1) Apr_3S(Φ)=Φ and Apr_3S(˜X)=˜X,

    (2) Apr_3S(F,E)(F,E),

    (3) (F,E)(G,E)Apr_3S(F,E)Apr_3S(G,E),

    (4) Apr_3S[(F,E)(G,E)]=Apr_3S(F,E)Apr_3S(G,E),

    (5) Apr_3S[(F,E)(G,E)]Apr_3S(F,E)Apr_3S(G,E),

    (6) Apr_3S(Apr_3S(F,E))=Apr_3S(F,E),

    (7) Apr_3S(F,E)=[¯Apr3S(F,E)c]c.

    Corollary 3.4. Let RE be a soft binary relation over X, (F,E)ˇSS(X)E and LE be a soft ideal on X. Then,

    Apr_1S(F,E)Apr_2S(F,E)Apr_3S(F,E)(F,E)¯Apr3S(F,E)¯Apr2S(F,E)¯Apr1S(F,E).

    Proof. Direct from Definitions 3.4–3.6, using Lemma 3.1.

    We dedicate this is the main section of the manuscript to display the properties of soft sets in the soft ideal approximation spaces. It has been generated using the concepts of R<x>R, soft neighborhoods and soft ideals. We study the main properties in soft ideal approximation spaces which are generalizations of the same properties of ideal approximation spaces given by Abbas et al. in [31] and provide various illustrative examples.

    Lemma 4.1. Let (X,RE,LE) is be a soft ideal approximation space. Then,

    (1) Apr_1S(<x>R)=<x>R,

    (2) Apr_2S(<x>R)=<x>R,

    (3) Apr_3S(R<x>R)=R<x>R.

    Proof. We will ensure that item (1) and the other items will be similar. From Theorem 3.3 part (3), it is clear that Apr_2S(<x>R)⊑<x>R.

    Conversely, we will ensure that <x>RApr_2S(<x>R). Let yˇ<x>R. Then, by Lemma 3.1 part(1), <y>R⊑<x>R. Thus, <y>R(<x>R)c=Φ. So, <y>R(<x>R)cˇLE. Hence, yˇApr_2S(<x>R). Thus, <x>RApr_2S(<x>R).

    Proposition 4.1. Let (X,RE,LE) be a soft ideal approximation space. For xyˇ˜X,

    (1) xˇ¯Apr1S(yE) iff <x>RyEΦ and xˇ¯Apr1S(yE) iff <x>RyE=Φ,

    (2) xˇ¯Apr2S(yE) iff <x>RyEˇLE and xˇ¯Apr2S(yE) iff <x>RyEˇLE,

    (3) xˇ¯Apr3S(yE) iff R<x>RyEˇLE and xˇ¯Apr3S(yE) iff R<x>RyEˇLE.

    Proof. We will prove the second statement and the others will be similar. Let xˇ¯Apr2S(yE). Then,

    xˇ[yE{zˇ˜X: <z>RyEˇLE}]. Thus, <x>RyEˇLE. Conversely, let <x>RyEˇLE. Then, by Definition 3.6, xˇ¯Apr2S(yE).

    Proposition 4.2. Let (X,RE,LE) be a soft ideal approximation space and <x>RˇLE. Then,

    (1) Apr_1S(xE)=xE=¯Apr1S(xE),

    (2) Apr_2S(xE)=xE=¯Apr2S(xE),

    (3) Apr_3S(xE)=xE=¯Apr3S(xE).

    Proof. We will prove that the second statement and the others will be similar. Let <x>RˇLE. Then, <x>R[xE]cˇLE. Thus, xˇApr_2S(xE). So, Apr_2S(xE)=xE. Also, <x>RˇLE induces that <x>RyEˇLE for all yˇX. Hence, ¯Apr2S(xE)=xE.

    Theorem 4.1. Let (X,RE,LE) be a soft ideal approximation space and xˇ˜X,(F,E)ˇSS(X)E.

    If <x>R(F,E)ˇLE, then

    (1) <x>R¯Apr1S(F,E)=Φ,

    (2) <x>R¯Apr2S(F,E)ˇLE,

    (3) R<x>R¯Apr3S(F,E)ˇLE.

    Proof. We will prove the second part and the others will be similar. Suppose <x>R(F,E)ˇLE. It is clear that [<x>RxE](F,E)ˇLE. Then, xˇDS(F,E). Thus, <x>RDS(F,E)=Φ. So,

    <x>RDS(F,E)ˇLE. Hence, [<x>R(F,E)DS(F,E)]ˇLE. Therefore,

    <x>R¯Apr2S(F,E)ˇLE.

    Definition 4.1. Let (X,RE,LE) be a soft ideal approximation space and (F,E)ˇSS(X)E. The soft exterior of (F,E) is ExtiS(F,E)=Apr_iS(F,E)c, i{1,2,3}.

    Lemma 4.2. Let (X,RE,LE) be a soft ideal approximation space and (F,E),(G,E)ˇSS(X)E. For i{1,2,3}, we have

    (1) ExtiS(Φ)=˜X and ExtiS(˜X)=Φ,

    (2) ExtiS(F,E)(F,E)c,

    (3) (F,E)(G,E)ExtiS(F,E)ExtiS(G,E),

    (4) ExtiS[(F,E)(G,E)]=ExtiS(G,E)ExtiS(F,E),

    (5) Apr_iS(F,E)=ExtiS[ExtiS(F,E)],

    (6) ExtiS(F,E)=ExtiS([ExtiS(F,E)]c).

    Proof. Straightforward from Theorems 3.2, 3.4, and 3.6.

    Definition 4.2. Let (X,RE,LE) be a soft ideal approximation space and (F,E)ˇSS(X)E. Then, a soft point xEˇSS(X)E is called:

    (i) A soft accumulation point of (F,E) if (<x>RxE)(F,E)Φ.

    The set of all soft ideal accumulation points of (F,E) is written as DS(F,E), that is,

    DS(F,E)={xEˇSS(X)E : (<x>RxE)(F,E)Φ}.

    (ii) A -soft ideal accumulation point of (F,E) if (<x>RxE)(F,E)ˇLE.

    The set of all -soft ideal accumulation points of (F,E) is written as DS(F,E), that is,

    DS(F,E)={xEˇSS(X)E : (<x>RxE)(F,E)ˇLE}.

    (iii) A -soft ideal accumulation point of (F,E) if (R<x>RxE)(F,E)ˇLE.

    The set of all -soft ideal accumulation points of (F,E) is written as DS(F,E), that is,

    DS(F,E)={xEˇSS(X)E : (R<x>RxE)(F,E)ˇLE}.

    Lemma 4.3. Let (X,RE,LE) be a soft ideal approximation space and (F,E)ˇSS(X)E. Then,

    (1) ¯Apr1S(F,E)=(F,E)DS(F,E),

    (2) ¯Apr1S(F,E)=(F,E) iff DS(F,E)(F,E),

    (3) ¯Apr2S(F,E)=(F,E)DS(F,E),

    (4) ¯Apr2S(F,E)=(F,E) iff DS(F,E)(F,E),

    (5) ¯Apr3S(F,E)=(F,E)DS(F,E),

    (6) ¯Apr3S(F,E)=(F,E) iff DS(F,E)(F,E).

    Proof. We will prove that the third and forth statements and the others will be similar.

    (3) Let xˇ¯Apr2S(F,E). Then, xˇ[(F,E){yEˇSS(X)E : <y>R(F,E)ˇLE}]. Then, we have either xˇ(F,E), that is,

    xˇ(F,E)DS(F,E) (4.1)

    or xˇ(F,E). So, xˇ{yEˇSS(X)E:<y>R(F,E)ˇLE}. In the latter case, we have (<x>RxE)(F,E)ˇLE. Hence, xˇDS(F,E), that is,

    xˇ(F,E)DS(F,E). (4.2)

    From Eqs (4.1) and (4.2), ¯Apr2S(F,E)(F,E)DS(F,E). Conversely, let xˇ(F,E)DS(F,E). Then, we have either xˇ(F,E), that is,

    xˇ¯Apr2S(F,E) (4.3)

    or xˇ(F,E). Thus, xˇDS(F,E). So (<x>RxE)(F,E)ˇLE. Hence, xˇ¯Apr2S(F,E), that is,

    xˇ¯Apr2S(F,E). (4.4)

    From Eqs (4.3) and (4.4), (F,E)DS(F,E)¯Apr2S(F,E).

    Therefore, ¯Apr2S(F,E)=(F,E)DS(F,E).

    (4) Let xˇ(F,E), that is, xˇ¯Apr2S(F,E). Then, <x>R(F,E)ˇLE. Thus,

    (<x>RxE)(F,E)ˇLE and xˇDS(F,E). Conversely, let DS(F,E)(F,E). Then, by part (1), DS(F,E)(F,E)=¯Apr2S(F,E)=(F,E).

    Lemma 4.4. Let (X,RE,LE) be a soft ideal approximation space and (F,E),(G,E)ˇSS(X)E. Then,

    (1) if (F,E)(G,E), then DS(F,E)DS(G,E) and DS(F,E)DS(G,E),

    (2) DS[(F,E)(G,E)]=DS(F,E)DS(F,E) and DS[(F,E)(G,E)]=DS(F,E)DS(F,E),

    (3) DS[(F,E)(G,E)]DS(F,E)DS(F,E) and DS[(F,E)(G,E)]DS(F,E)DS(F,E),

    (4) DS[(F,E)DS(F,E)](F,E)DS(F,E) and DS[(F,E)DS(F,E)](F,E)DS(F,E).

    Proof. (1) Suppose (F,E)(G,E) and let xˇDS(F,E). Then, [<x>RxE](F,E)ˇLE. Thus, [<x>RxE](G,E)ˇLE. So, xˇDS(G,E). The second part is easily proved.

    (2) Since (F,E)(F,E)(G,E) and (G,E)(F,E)(G,E), by part (1), we have DS(F,E)DS(G,E)DS(F,E)(G,E)).

    Conversely, let xˇ(DS(F,E)DS(G,E). Then, xˇDS(F,E) and xˇDS(G,E). Thus, (<x>RxE)(F,E)ˇLE and (<x>RxE)(G,E)ˇLE. So, (<x>RxE)(F,E)(G,E))ˇLE. Hence, xˇDS[(F,E)(G,E)]. The proof of the second part is similar.

    (3) Similar to part (2).

    (4) Let xˇ(F,E)DS(F,E). It is obvious that xˇ(F,E) and (<x>RxE)(F,E)ˇLE. Then, <x>R(F,E)ˇLE. Thus, xˇ¯Apr2S(F,E). So, xˇ¯Apr2S(¯Apr2S(F,E)). Hence, xˇDS(¯Apr2S(F,E))=DS(F,E)DS(F,E)). Therefore, DS(F,E)DS(F,E)(F,E)DS(F,E). The proof of the second part is similar.

    Corollary 4.1. Let (X,RE,LE) be any soft ideal approximation space and (F,E)ˇSS(X)E. Then,

    DS(F,E)DS(F,E)DS(F,E).

    Proof. Let xˇDS(F,E). Then, (<x>RxE)(F,E)=Φ. Thus, (<x>RxE)(F,E)ˇLE. So, xˇDS(F,E) and (R<x>RxE)(F,E)ˇLE, where R<x>R⊑<x>R. Hence, xˇDS(F,E). Therefore, DS(F,E)DS(F,E)DS(F,E).

    Remark 4.1. The converse of the previous result is not true.

    Example 4.1. Let X={a,b,c} associated with a set of parameters E={e1,e2}. Let RE be a soft relation of X and LE be a soft ideal on X, defined respectively by:

    R={(e1,{(a,a),(a,b),(a,c),(b,b),(b,c),(c,c)}),(e2,{(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,b),
    (c,c)})}

    LE={Φ,(F1,E),(F2,E),(F3,E)} where,

    (F1,E)={(e1,{c}),(e2,ϕ)},(F2,E)={(e1,ϕ),(e2,{c})},(F3,E)={(e1,{c}),(e2,{c})}.

    Then, <a>R={(e1,{a,b,c}),(e2,{a,b,c})},<b>R={(e1,{b,c}),(e2,{b,c})},

    <c>R=cE. Also, R<a>=aE,R<b>={(e1,{a,b}),(e2,{a,b})},R<c>=<a>R. Thus, R<a>R=aE,R<b>R=bE,R<c>R=cE. Suppose (F,E)={(e1,{b,c}),(e2,{b,c})}. Then, we have:

    (<a>RaE)(F,E)=(F,E)Φ,
    (<b>RbE)(F,E)=cEΦ,
    (<c>RcE)(F,E)=Φ.

    Thus, aˇDS(F,E),bˇDS(F,E),cˇDS(F,E). So, DS(F,E)={(e1,{a,b}),(e2,{a,b})}. On the other hand, we get:

    (<a>RaE)(F,E)=(F,E)ˇLE,
    (<b>RbE)(F,E)=(F3,E)ˇLE,
    (<c>RcE)(F,E)=ΦˇLE.

    Thus, aˇDS(F,E),bˇDS(F,E),cˇDS(F,E). Hence, DS(F,E)=aE. Also, we have:

    (R<a>RaE)(F,E)=ΦˇLE,
    (R<b>RbE)(F,E)=ΦˇLE,
    (R<c>RcE)(F,E)=ΦˇLE.

    Then, aˇDS(F,E),bˇDS(F,E),cˇDS(F,E). Thus, DS(F,E)=Φ. So, DS(F,E)

    Definition 4.3. Let be any soft ideal approximation space and . Then, is said to be:

    (i) soft dense if

    (ii) -soft ideal dense if

    (iii) -soft ideal dense if

    (iv) soft nowhere dense if

    (v) -soft ideal nowhere dense if

    (vi) -soft ideal nowhere dense if

    Corollary 4.2. Let be any soft ideal approximation space and . Then,

    (1) -soft ideal dense -soft ideal dense soft dense,

    (2) soft nowhere dense -soft ideal nowhere dense -soft ideal nowhere dense.

    Proof. Immediately from Definition 4.3 and part (3) of Theorem 3.5.

    Example 4.2. Let ,

    and , where

    Therefore, we have Also, Thus, Suppose Then, Also, Hence, is -soft ideal dense but not -soft ideal dense.

    Corollary 4.3. Let be any soft ideal approximation space and . Then,

    (1) If is soft dense, then is soft nowhere dense.

    (2) If is -soft ideal dense, then is -soft ideal nowhere dense.

    (3) If is -soft ideal dense, then is -nowhere dense.

    Proof. Suppose is soft dense. Then, . Thus, and

    So, Hence,

    is nowhere soft dense.

    Suppose is -soft ideal dense. Then, Thus, So,

    and Hence,

    is -soft ideal nowhere dense.

    Similar to part (2).

    In this section, we introduce soft lower separation axioms via soft binary relations and soft ideal as a generalization of lower separation axioms given in [31]. We scrutinize its essential characterizations and infer some of its relationships associated with the soft ideal closure operators. Some illustrative examples are given. In an approximation space where is an equivalence relation on , a general topology is generated by the lower approximations or the upper approximations of any subset as follows. or . In the soft case, it is an extension of the same definitions.

    Definition 5.1. A soft approximation space is said to be a soft- space if , there exists such that

    A soft ideal approximation space is said to be a soft- space if , there exists such that

    A soft ideal approximation space is said to be a soft- space if , there exists such that

    Proposition 5.1. For a soft ideal approximation space , these properties are equivalent:

    (1) is a soft- space.

    (2) for all

    Proof.

    : For each by part (1), there exists such that , Thus, So, and by Proposition 4.1 part (1), Similarly, we can prove that Therefore,

    : Suppose part (2) holds and let Then, or By Proposition 4.1 part (2), or Thus, or Therefore, is soft- space.

    Corollary 5.1. For a soft approximation space , these properties are equivalent:

    is a soft- space.

    for each

    Corollary 5.2. For a soft ideal approximation space , these properties are equivalent:

    (1) is a soft- space.

    (2) for all

    Definition 5.2. A soft approximation space is said to be a soft- space if , there exist such that

    A soft ideal approximation space is said to be a soft- space if , there exist such that

    A soft ideal approximation space is said to be a soft- space if , there exist such that

    Proposition 5.2. For a soft ideal approximation space , these properties are equivalent:

    (1) is a soft- space.

    (2) for all

    (3) for each

    Proof. : Suppose is a soft- space and let . Thus, for and such that Thus, So, that is, Hence,

    : Suppose part (2) holds and let Then, but Thus,

    : Suppose part (3) holds and By part (3), Thus, and that is, and So, there exist and such that Therefore, is a soft- space.

    Corollary 5.3. For a soft approximation space , these properties are equivalent:

    (1) is a soft- space.

    (2) for all

    (3) for each

    Corollary 5.4. For a soft ideal approximation space , these properties are equivalent:

    (1) is a soft- space.

    (2) for all

    (3) for each

    Definition 5.3. A soft approximation space is said to be a soft- space if,

    for all ,

    A soft ideal approximation space is said to be a soft- space if, for all

    A soft ideal approximation space is said to be a soft- space if, for all

    Proposition 5.3. For a soft ideal approximation space , these properties are equivalent:

    (1) is a soft- space,

    (2) if , then for all

    Proof.

    (1) : Suppose statement (1) holds, and let be two soft points in . Then, or

    If , then and

    If then and Thus, and So, and Hence, in either case, statement (2) holds.

    (2) : Suppose that statement (2) holds and let Then, we have

    If then

    (5.1)

    If then

    (5.2)

    From (5.1) and (5.2), the proof is complete.

    Corollary 5.5. For a soft approximation space , these properties are equivalent:

    (1) is a soft- space,

    (2) if , then for any

    Corollary 5.6. For a soft ideal approximation space , these properties are equivalent:

    (1) is a soft- space,

    (2) if , then for all

    Definition 5.4. A soft approximation space is said to be a soft- space if , there exist such that

    A soft ideal approximation space is said to be a soft- space if , there exist such that

    A soft ideal approximation space is said to be a soft- space if , there exist such that

    Theorem 5.1. For a soft ideal approximation space , these properties are equivalent:

    (1) is a soft- space,

    (2) for any

    Proof. : Suppose is a soft- space and let . Then, there exist

    such that and Thus, and So that is, Hence, and that is, Therefore,

    : Suppose part (2) holds and let Then, there exists such that . Let . Then, (from Theorem 3.3 part (7)) and so Also, Hence, is a soft- space.

    Corollary 5.7. For a soft approximation space , these properties are equivalent:

    (1) is a soft- space,

    (2) for all

    Corollary 5.8. For a soft ideal approximation space , these properties are equivalent:

    (1) is a soft- space,

    (2) for all

    Corollary 5.9. For a soft ideal approximation space , these conditions hold:

    (1) soft- = soft- + soft-,

    (2) soft- = soft- + soft-,

    (3) soft- = soft- + soft-.

    Proof. Straightforward from Definition 5.3, Propositions 5.1 and 5.2, and Corollaries 5.1–5.4.

    Remark 5.1. From Definitions 5.1, 5.2, and 5.4 we have the following implication.

    Example 5.1 (1) Let with and be a soft relation over defined by and Thus,

    Thus, , we have:

    So, is a soft- space. But is not a soft- space, since if and then and which is impossible because is infinite soft set and is finite soft set.

    (2) From part (1), we have

    Then, , we have:

    Hence, is soft- and soft-. However, is neither soft- space nor soft-. By the same way, any one can add examples to show that the above implication is not reversible.

    Definition 5.5. Let and be two soft approximation spaces and let a soft ideal on . Then,

    a function is said to be soft continuous if , that is, for all

    A function is said to be -soft continuous if , that is, for all

    A function is said to be -soft continuous if , that is, for all

    Remark 5.2. From Corollary 3.4, we have the following implications:

    Example 5.2. Let associated with the parameters . Let be a soft relation of , and be a soft ideal on , defined respectively by:

    where

    Then, Also, Thus, On the other hand, let associated with the parameters . Let be a soft relation over defined by:

    Then, Also, Thus,

    Now, define the function where is a function defined by and is a function defined by By calculating and of a soft set , it is clear that is * -soft continuous. However, is not soft continuous, where

    Theorem 5.2. Let be an injective soft continuous function. Then,

    is a soft -space if is a soft- space for

    Proof. Suppose is a soft- space for and let in . For since is injective, . Then, by the hypothesis, there exist such that and that is, and

    Since is soft continuous, Thus, that is there exist

    such that and So, is a soft space. For the proofs are similar.

    Corollary 5.10. Let be an injective soft continuous function. Thus, is a soft -space if is a soft- space for

    Corollary 5.11. Let be an injective soft continuous function. Then, is a soft -space if is a soft- space for

    In this section, We reformulate and study soft connectedness in [31] with respect to these soft ideal approximation spaces. Some examples are submitted to explain the definitions.

    Definition 6.1. Let be a soft approximation space. Then,

    are called soft separated sets if

    is said to be a soft disconnected set if there exist soft separated sets such that . is said to be soft connected if it is not soft disconnected.

    is said to be a soft disconnected space if there exist soft separated sets such that . is said to be a soft connected space if it is not soft disconnected space.

    Definition 6.2. Let be a soft ideal approximation space. Then,

    are called - soft separated (resp. - soft separated) sets if (resp. ).

    is called a -soft disconnected (resp. -soft disconnected) set if there exist -soft separated (resp. -soft separated) sets such that . is said to be -soft connected (resp. -soft connected) if it is not -soft disconnected (resp. -soft disconnected).

    is called a -soft disconnected (resp. -soft disconnected) space if there exist -soft separated (resp. -soft separated) sets such that . is called a -soft connected (resp. -soft connected) space if it is not a -soft disconnected (resp. -soft disconnected) space.

    Remark 6.1. The following implications are correct:

    and so

    Example 6.1. Let associated with a set of parameters . Let be a soft relation over defined by:

    Then, Also, Thus,

    (1) Let be a soft ideal on defined by:

    where

    Then, we have

    Thus, is a soft connected space. However, we get

    So, is not a -soft connected space.

    (2) Consider where

    Then, we get

    Thus, is a -soft connected space. However, we have

    So, is not a -soft connected space.

    Proposition 6.1. Let be a soft ideal approximation space. Then, these properties are equivalent:

    (1) is -soft connected,

    (2) for each with and or

    (3) for each with and or

    Proof. (1) : Suppose part (1) holds and let with ,

    such that and Then,

    Thus, So, are -soft separated sets. Since , or by part (1).

    (2) and Clear.

    Corollary 6.1. Let be a soft approximation space. Then, these properties are equivalent:

    (1) is soft connected,

    (2) for each with and or

    (3) for each with and or

    Corollary 6.2. Let be a soft ideal approximation space. Then, these properties are equivalent:

    (1) is -soft connected.

    (2) For each with and or

    (3) For each with and or

    Theorem 6.1. Let be a soft ideal approximation space and be -soft connected. If are -soft separated sets with then either or

    Proof. Suppose are -soft separated sets with Then, we have

    On the other hand, we get

    Also,

    Thus, and are -soft separated sets with However, is -soft connected, which implies that or

    Corollary 6.3. Let be a soft approximation space and be soft connected. If are soft separated sets with then either or

    Corollary 6.4. Let be a soft ideal approximation space and be -soft connected. If are -soft separated sets with then either or

    Theorem 6.2. Let be a -soft continuous function. Then,

    is a soft connected set if is -soft connected.

    Proof. Assume that is -soft connected in Suppose that is soft disconnected. Thus, there exist two soft separated sets with that is, Since is -soft continuous, Thus, we have

    Also, we have

    So, and are -soft separated sets in , that is,

    Hence, is -soft disconnected, which contradicts that is -soft connected. Therefore, is a soft connected set in .

    Corollary 6.5. Let be a soft continuous function. Then, is soft connected set, if is soft connected.

    Corollary 6.6. Let be a -soft continuous function. Then,

    is soft connected set if is -soft connected.

    Herein, we first compare the current purposed methods in Definitions 3.4–3.6 and demonstrate that the method given in Definition 3.6 is the best in terms of developing the soft approximation operators and the values of soft accuracy. Then, we clarify that the third approach in Definition 3.6 produces soft accuracy measures of soft subsets higher than their counterparts displayed in the previous method 2.4 in [17]. Moreover, we applied these approaches to handle real-life problems.

    Definition 7.1. Let be a soft ideal approximation space. Then, the soft boundary region of a soft set and the soft accuracy measure of an absolute soft set with respect to the soft binary relation are defined respectively by:

    where Note that denotes the cardinality of set

    Proposition 7.1. Let be a soft ideal approximation space and Then,

    (1)

    (2)

    Proof. (1) Let Then, from Corollary 3.4, we have

    . Again, by Corollary 3.4,

    if then Hence,

    (2) From Corollary 3.4, we have

    Proposition 7.2. Let and be soft ideal approximation spaces such that Thus, for each we have

    (1)

    (2)

    (3)

    (4)

    Proof.

    (1) Let Then, Since Thus, Therefore, Hence,

    (2) Let Then, Since Thus, Therefore, Hence,

    (3), (4): It is immediately obtained by parts (1) and (2).

    Corollary 7.1. Let and be soft ideal approximation spaces such that Thus, for each we have

    (1)

    (2)

    (3)

    (4)

    Remark 7.1. Proposition 7.2 shows that the soft boundary region of a soft set decreases as the soft ideal increases as illustrated in the next example.

    Example 7.1. Let associated with a set of parameters . Let be a soft relation over . Let be soft ideals on , defined respectively by:

    Therefore,

    Let Then,

    Also

    It is clear that

    Remark 7.2. From Proposition 5.2, one can deduce that Definition 3.6 improves the soft boundary region which means decreasing for a soft set , and improves the soft accuracy measure which means increasing for that soft set by increasing the soft lower approximation and decreasing the soft upper approximation in comparison to the methods in Definitions 3.4, 3.5, and Definition 2.4 in [17]. So, the suggested method in Definition 3.6 is more accurate in decision-making. As a special case:

    If is soft symmetric relation, then the soft approximations in Definition 3.6 coincide with the soft approximations in Definition 3.5.

    If and is soft symmetric relation, then the soft approximations in Definition 3.5 coincide with the soft approximations in Definition 3.5.

    If and is soft reflexive and soft transitive relation, then the soft approximations in Definition 3.6 coincide with the previous soft approximations in [17].

    Example 7.2. Selection of a house:

    Considering is a collection of six houses where {expensive, beautiful, cheap, in green surroundings, wooden modern, in good repair, in bad repair} be a set of parameters.

    Suppose Mr.Z wants to purchase a house on the following parametric set {beautiful, cheap, in green surroundings, wooden, in good repair}. Consider

    Define a soft equivalence relation . The soft equivalence classes for each are obtained as follows:

    Therefore, Consider is a soft set over be a soft ideal over . The soft representation of the equivalence relation is explained in Table 1. In Table 2, the soft approximations, soft boundary region, and soft accuracy measure of a soft set by using our suggested method in Definition 3.6. This method is the best tool to help Mr.Z in his decision-making about selecting the house that is most suitable to his choice of parameters. For example, take then from Table 2, the soft lower and soft upper approximations, soft boundary region, and soft accuracy measure are , and , respectively. One can see that Mr.Z will decide to buy the house according to his choice parameters in .

    Table 1.  Soft equivalence relation representation of houses under consideration.
    1 1 1 1 1
    0 1 1 0 1
    1 0 0 1 1
    0 1 1 0 1
    0 1 1 0 1
    0 0 1 1 1
    0 1 1 0 1
    1 1 1 1 1
    0 0 0 0 1
    1 1 1 1 1
    1 1 1 1 1
    1 0 1 0 1
    1 0 0 1 1
    0 0 0 0 1
    1 1 1 1 1
    0 0 0 0 1
    0 0 0 0 1
    0 0 0 1 1
    0 1 1 0 1
    1 1 1 1 1
    0 0 0 0 1
    1 1 1 1 1
    1 1 1 1 1
    1 0 1 0 1
    0 1 1 0 1
    1 1 1 1 1
    0 0 0 0 1
    1 1 1 1 1
    1 1 1 1 1
    1 0 1 0 1
    0 0 1 1 1
    1 0 1 0 1
    0 0 0 1 1
    1 0 1 0 1
    1 0 1 0 1
    1 1 1 1 1

     | Show Table
    DownLoad: CSV
    Table 2.  Soft approximations, soft boundary region and soft accuracy measure of a soft set of Definition 3.6.

     | Show Table
    DownLoad: CSV

    Example 7.3. Selection of a car:

    Suppose a person Mr.Z wants to buy a car from the alternatives Let be the universe of ten different cars and let be the set of attributes, where refers to price, refers to color, and refers to car brands.

    The parameters are characterized as follows:

    The price of a car includes under 30 lacs, between 31 and 35 lacs, and between 36 and 40 lacs.

    The car brand includes Honda Accord, Audi, Mercedes Benz, and BMW.

    The color of a car includes black, white, and silver.

    Define a soft equivalence relation for each which describes the advantages of the car for which the person Mr.Z will buy. The soft equivalence classes for each are obtained as follows:

    which means that the price of cars and is under 30 lacs; the price of cars , and is between 31 and 35 lacs; and the price of cars , and is between 36 and 40 lacs.

    which represents that the brand of car is Honda Accord; the brand of car is Audi; the brand of cars and is Mercedes Benz; and the brand of car is BMW. For : are which represents that the color of cars and is black; the color of car is white; and the color of car is silver.

    Therefore,

    Consequently, anyone can offer a soft ideal to extend an example similar to the one in Table 2 to help Mr.Z in his decision-making about selecting the car that is most suitable according to the given parameters.

    For example, let be a soft ideal over and consisting of these cars which are most acceptable for Mr.Z. Thus,

    and Mr.Z will buy the car which is under 30 lacs, a Honda Accord, and is white.

    This paper introduced new soft closure operators based on soft ideals, defining soft topological spaces. To that end, soft accumulation points, soft subspaces, and soft lower separation axioms of such spaces are defined and studied. Moreover, soft connectedness in these spaces is defined, which enables us to make more generalizations and studies. The obtained results are newly presented and could enrich soft topology theory. Finally, applications in multi criteria group decision making by using our methods to present the importance of our soft ideals approximations have been presented.

    As it is well-known that the soft interior and soft closure topological operators behave similarly to the lower and upper soft approximations. So, in forthcoming works, we plan to study the counterparts of these models via topological structures. In addition, we will benefit from the hybridization of rough set theory with some approaches, such as fuzzy sets and soft fuzzy sets, to introduce these approximation spaces via these hybridized frames and show their role in efficiently dealing with uncertain knowledge.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors extend their appreciation to the Deputy-ship for Research and Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number ISP-2024.

    The authors declare that they have no conflicts of interest.


    Acknowledgments



    The authors would like to acknowledge the financial support from Australian Research Council (Grant No. DP160102491).

    Conflict of interest



    The authors declare no conflict of interests.

    [1] Hu C, Ashok D, Nisbet DR, et al. (2019) Bioinspired surface modification of orthopedic implants for bone tissue engineering. Biomaterials 119366. doi: 10.1016/j.biomaterials.2019.119366
    [2] Karsenty G, Olson EN (2016) Bone and muscle endocrine functions: unexpected paradigms of inter-organ communication. Cell 164: 1248-1256. doi: 10.1016/j.cell.2016.02.043
    [3] Rossi M, Battafarano G, Pepe J, et al. (2019) The endocrine function of osteocalcin regulated by bone resorption: A lesson from reduced and increased bone mass diseases. Int J Mol Sci 20: 4502. doi: 10.3390/ijms20184502
    [4] Loebel C, Burdick JA (2018) Engineering stem and stromal cell therapies for musculoskeletal tissue repair. Cell Stem Cell 22: 325-339. doi: 10.1016/j.stem.2018.01.014
    [5] Dimitriou R, Jones E, McGonagle D, et al. (2011) Bone regeneration: current concepts and future directions. BMC Med 9: 66. doi: 10.1186/1741-7015-9-66
    [6] Nordin M, Frankel VH (2001)  Basic Biomechanics of the Musculoskeletal System, 3 Eds USA: Lippincott Williams & Wilkins.
    [7] Kobayashi S, Takahashi HE, Ito A, et al. (2003) Trabecular minimodeling in human iliac bone. Bone 32: 163-169. doi: 10.1016/S8756-3282(02)00947-X
    [8] Bartl R, Bartl C (2019) Control and regulation of bone remodelling. The Osteoporosis Manual Cham: Springer, 31-39. doi: 10.1007/978-3-030-00731-7_4
    [9] Kenkre JS, Bassett JHD (2018) The bone remodelling cycle. Ann Clin Biochem 55: 308-327. doi: 10.1177/0004563218759371
    [10] Prendergast PJ, Huiskes R (1995) The biomechanics of Wolff's law: recent advances. Irish J Med Sci 164: 152-154. doi: 10.1007/BF02973285
    [11] Wegst UGK, Bai H, Saiz E, et al. (2015) Bioinspired structural materials. Nat Mater 14: 23-36. doi: 10.1038/nmat4089
    [12] Reznikov N, Shahar R, Weiner S (2014) Bone hierarchical structure in three dimensions. Acta Biomater 10: 3815-3826. doi: 10.1016/j.actbio.2014.05.024
    [13] Weiner S, Wagner HD (1998) The material bone: structure-mechanical function relations. Annu Rev Mater Sci 28: 271-298. doi: 10.1146/annurev.matsci.28.1.271
    [14] Recker RR, Kimmel DB, Dempster D, et al. (2011) Issues in modern bone histomorphometry. Bone 49: 955-964. doi: 10.1016/j.bone.2011.07.017
    [15] Eriksen EF, Vesterby A, Kassem M, et al. (1993) Bone remodeling and bone structure. Physiology and Pharmacology of Bone Heidelberg: Springer, 67-109. doi: 10.1007/978-3-642-77991-6_2
    [16] Augat P, Schorlemmer S (2006) The role of cortical bone and its microstructure in bone strength. Age Ageing 35: ii27-ii31. doi: 10.1093/ageing/afl081
    [17] Kozielski M, Buchwald T, Szybowicz M, et al. (2011) Determination of composition and structure of spongy bone tissue in human head of femur by Raman spectral mapping. J Mater Sci: Mater Med 22: 1653-1661. doi: 10.1007/s10856-011-4353-0
    [18] Cross LM, Thakur A, Jalili NA, et al. (2016) Nanoengineered biomaterials for repair and regeneration of orthopedic tissue interfaces. Acta Biomater 42: 2-17. doi: 10.1016/j.actbio.2016.06.023
    [19] Zebaze R, Seeman E (2015) Cortical bone: a challenging geography. J Bone Miner Res 30: 24-29. doi: 10.1002/jbmr.2419
    [20] Liu Y, Luo D, Wang T (2016) Hierarchical structures of bone and bioinspired bone tissue engineering. Small 12: 4611-4632. doi: 10.1002/smll.201600626
    [21] Brodsky B, Persikov AV (2005) Molecular structure of the collagen triple helix. Adv Protein Chem 70: 301-339. doi: 10.1016/S0065-3233(05)70009-7
    [22] Cui FZ, Li Y, Ge J (2007) Self-assembly of mineralized collagen composites. Mater Sci Eng R Rep 57: 1-27. doi: 10.1016/j.mser.2007.04.001
    [23] Wang Y, Azaïs T, Robin M, et al. (2012) The predominant role of collagen in the nucleation, growth, structure and orientation of bone apatite. Nat Mater 11: 724-733. doi: 10.1038/nmat3362
    [24] Bentmann A, Kawelke N, Moss D, et al. (2010) Circulating fibronectin affects bone matrix, whereas osteoblast fibronectin modulates osteoblast function. J Bone Miner Res 25: 706-715.
    [25] Szweras M, Liu D, Partridge EA, et al. (2002) α2-HS glycoprotein/fetuin, a transforming growth factor-β/bone morphogenetic protein antagonist, regulates postnatal bone growth and remodeling. J Biol Chem 277: 19991-19997. doi: 10.1074/jbc.M112234200
    [26] Boskey AL, Robey PG (2013) The regulatory role of matrix proteins in mineralization of bone. Osteoporosis, 4 Eds Academic Press, 235-255. doi: 10.1016/B978-0-12-415853-5.00011-X
    [27] Boskey AL (2013) Bone composition: relationship to bone fragility and antiosteoporotic drug effects. Bonekey Rep 2: 447. doi: 10.1038/bonekey.2013.181
    [28] Stock SR (2015) The mineral–collagen interface in bone. Calcified Tissue Int 97: 262-280. doi: 10.1007/s00223-015-9984-6
    [29] Nikel O, Laurencin D, McCallum SA, et al. (2013) NMR investigation of the role of osteocalcin and osteopontin at the organic–inorganic interface in bone. Langmuir 29: 13873-13882. doi: 10.1021/la403203w
    [30] He G, Dahl T, Veis A, et al. (2003) Nucleation of apatite crystals in vitro by self-assembled dentin matrix protein 1. Nat Mater 2: 552-558. doi: 10.1038/nmat945
    [31] Clarke B (2008) Normal bone anatomy and physiology. Clin J Am Soc Nephro 3: S131-S139. doi: 10.2215/CJN.04151206
    [32] Olszta MJ, Cheng X, Jee SS, et al. (2007) Bone structure and formation: A new perspective. Mater Sci Eng R Rep 58: 77-116. doi: 10.1016/j.mser.2007.05.001
    [33] Nair AK, Gautieri A, Chang SW, et al. (2013) Molecular mechanics of mineralized collagen fibrils in bone. Nature Commun 4: 1724. doi: 10.1038/ncomms2720
    [34] Landis WJ (1995) The strength of a calcified tissue depends in part on the molecular structure and organization of its constituent mineral crystals in their organic matrix. Bone 16: 533-544. doi: 10.1016/8756-3282(95)00076-P
    [35] Hunter GK, Hauschka PV, POOLE RA, et al. (1996) Nucleation and inhibition of hydroxyapatite formation by mineralized tissue proteins. Biochem J 317: 59-64. doi: 10.1042/bj3170059
    [36] Oikeh I, Sakkas P, Blake D P, et al. (2019) Interactions between dietary calcium and phosphorus level, and vitamin D source on bone mineralization, performance, and intestinal morphology of coccidia-infected broilers. Poult Sci 11: 5679-5690. doi: 10.3382/ps/pez350
    [37] Boyce BF, Rosenberg E, de Papp AE, et al. (2012) The osteoclast, bone remodelling and treatment of metabolic bone disease. Eur J Clin Invest 42: 1332-1341. doi: 10.1111/j.1365-2362.2012.02717.x
    [38] Teitelbaum SL (2000) Bone resorption by osteoclasts. Science 289: 1504-1508. doi: 10.1126/science.289.5484.1504
    [39] Yoshida H, Hayashi SI, Kunisada T, et al. (1990) The murine mutation osteopetrosis is in the coding region of the macrophage colony stimulating factor gene. Nature 345: 442-444. doi: 10.1038/345442a0
    [40] Roodman GD (2006) Regulation of osteoclast differentiation. Ann NY Acad Sci 1068: 100-109. doi: 10.1196/annals.1346.013
    [41] Martin TJ (2013) Historically significant events in the discovery of RANK/RANKL/OPG. World J Orthop 4: 186-197. doi: 10.5312/wjo.v4.i4.186
    [42] Coetzee M, Haag M, Kruger MC (2007) Effects of arachidonic acid, docosahexaenoic acid, prostaglandin E2 and parathyroid hormone on osteoprotegerin and RANKL secretion by MC3T3-E1 osteoblast-like cells. J Nutr Biochem 18: 54-63. doi: 10.1016/j.jnutbio.2006.03.002
    [43] Steeve KT, Marc P, Sandrine T, et al. (2004) IL-6, RANKL, TNF-alpha/IL-1: interrelations in bone resorption pathophysiology. Cytokine Growth F R 15: 49-60. doi: 10.1016/j.cytogfr.2003.10.005
    [44] Mellis DJ, Itzstein C, Helfrich M, et al. (2011) The skeleton: a multi-functional complex organ. The role of key signalling pathways in osteoclast differentiation and in bone resorption. J Endocrinol 211: 131-143. doi: 10.1530/JOE-11-0212
    [45] Silva I, Branco J (2011) Rank/Rankl/opg: literature review. Acta Reumatol Port 36: 209-218.
    [46] Martin TJ, Sims NA (2015) RANKL/OPG; Critical role in bone physiology. Rev Endocr Metab Dis 16: 131-139. doi: 10.1007/s11154-014-9308-6
    [47] Wang Y, Qin QH (2012) A theoretical study of bone remodelling under PEMF at cellular level. Comput Method Biomec 15: 885-897. doi: 10.1080/10255842.2011.565752
    [48] Weitzmann MN, Pacifici R (2007) T cells: unexpected players in the bone loss induced by estrogen deficiency and in basal bone homeostasis. Ann NY Acad Sci 1116: 360-375. doi: 10.1196/annals.1402.068
    [49] Duong LT, Lakkakorpi P, Nakamura I, et al. (2000) Integrins and signaling in osteoclast function. Matrix Biol 19: 97-105. doi: 10.1016/S0945-053X(00)00051-2
    [50] Stenbeck G (2002) Formation and function of the ruffled border in osteoclasts. Semin Cell Dev Biol 13: 285-292. doi: 10.1016/S1084952102000587
    [51] Jurdic P, Saltel F, Chabadel A, et al. (2006) Podosome and sealing zone: specificity of the osteoclast model. Eur J Cell Biol 85: 195-202. doi: 10.1016/j.ejcb.2005.09.008
    [52] Väänänen HK, Laitala-Leinonen T (2008) Osteoclast lineage and function. Arch Biochem Biophys 473: 132-138. doi: 10.1016/j.abb.2008.03.037
    [53] Vaananen HK, Zhao H, Mulari M, et al. (2000) The cell biology of osteoclast function. J cell Sci 113: 377-381.
    [54] Sabolová V, Brinek A, Sládek V (2018) The effect of hydrochloric acid on microstructure of porcine (Sus scrofa domesticus) cortical bone tissue. Forensic Sci Int 291: 260-271. doi: 10.1016/j.forsciint.2018.08.030
    [55] Delaissé JM, Engsig MT, Everts V, et al. (2000) Proteinases in bone resorption: obvious and less obvious roles. Clin Chim Acta 291: 223-234. doi: 10.1016/S0009-8981(99)00230-2
    [56] Logar DB, Komadina R, Preželj J, et al. (2007) Expression of bone resorption genes in osteoarthritis and in osteoporosis. J Bone Miner Metab 25: 219-225. doi: 10.1007/s00774-007-0753-0
    [57] Lorget F, Kamel S, Mentaverri R, et al. (2000) High extracellular calcium concentrations directly stimulate osteoclast apoptosis. Biochem Bioph Res Co 268: 899-903. doi: 10.1006/bbrc.2000.2229
    [58] Nesbitt SA, Horton MA (1997) Trafficking of matrix collagens through bone-resorbing osteoclasts. Science 276: 266-269. doi: 10.1126/science.276.5310.266
    [59] Xing L, Boyce BF (2005) Regulation of apoptosis in osteoclasts and osteoblastic cells. Biochem Bioph Res Co 328: 709-720. doi: 10.1016/j.bbrc.2004.11.072
    [60] Hughes DE, Wright KR, Uy HL, et al. (1995) Bisphosphonates promote apoptosis in murine osteoclasts in vitro and in vivo. J Bone Miner Res 10: 1478-1487. doi: 10.1002/jbmr.5650101008
    [61] Choi Y, Arron JR, Townsend MJ (2009) Promising bone-related therapeutic targets for rheumatoid arthritis. Nat Rev Rheumatol 5: 543-548. doi: 10.1038/nrrheum.2009.175
    [62] Harvey NC, McCloskey E, Kanis JA, et al. (2017) Bisphosphonates in osteoporosis: NICE and easy? Lancet 390: 2243-2244. doi: 10.1016/S0140-6736(17)32850-7
    [63] Ducy P, Schinke T, Karsenty G (2000) The osteoblast: a sophisticated fibroblast under central surveillance. Science 289: 1501-1504. doi: 10.1126/science.289.5484.1501
    [64] Katagiri T, Takahashi N (2002) Regulatory mechanisms of osteoblast and osteoclast differentiation. Oral dis 8: 147-159. doi: 10.1034/j.1601-0825.2002.01829.x
    [65] Kretzschmar M, Liu F, Hata A, et al. (1997) The TGF-beta family mediator Smad1 is phosphorylated directly and activated functionally by the BMP receptor kinase. Gene Dev 11: 984-995. doi: 10.1101/gad.11.8.984
    [66] Bennett CN, Longo KA, Wright WS, et al. (2005) Regulation of osteoblastogenesis and bone mass by Wnt10b. P Natl A Sci 102: 3324-3329. doi: 10.1073/pnas.0408742102
    [67] Wang Y, Qin QH, Kalyanasundaram S (2009) A theoretical model for simulating effect of parathyroid hormone on bone metabolism at cellular level. Mol Cell Biomech 6: 101-112.
    [68] Elefteriou F, Ahn JD, Takeda S, et al. (2005) Leptin regulation of bone resorption by the sympathetic nervous system and CART. Nature 434: 514-520. doi: 10.1038/nature03398
    [69] Proff P, Römer P (2009) The molecular mechanism behind bone remodelling: a review. Clin Oral Invest 13: 355-362. doi: 10.1007/s00784-009-0268-2
    [70] Katsimbri P (2017) The biology of normal bone remodelling. Eur J Cancer Care 26: e12740. doi: 10.1111/ecc.12740
    [71] Fratzl P, Weinkamer R (2007) Nature's hierarchical materials. Prog Mater Sci 52: 1263-1334. doi: 10.1016/j.pmatsci.2007.06.001
    [72] Athanasiou KA, Zhu CF, Lanctot DR, et al. (2000) Fundamentals of biomechanics in tissue engineering of bone. Tissue Eng 6: 361-381. doi: 10.1089/107632700418083
    [73] Takahashi N, Udagawa N, Suda T (1999) A new member of tumor necrosis factor ligand family, ODF/OPGL/TRANCE/RANKL, regulates osteoclast differentiation and function. Biocheml Bioph Res Co 256: 449-455. doi: 10.1006/bbrc.1999.0252
    [74] Nakashima T, Hayashi M, Fukunaga T, et al. (2011) Evidence for osteocyte regulation of bone homeostasis through RANKL expression. Nat Med 17: 1231-1234. doi: 10.1038/nm.2452
    [75] Prideaux M, Findlay DM, Atkins GJ (2016) Osteocytes: the master cells in bone remodelling. Curr Opin Pharmacol 28: 24-30. doi: 10.1016/j.coph.2016.02.003
    [76] Dallas SL, Prideaux M, Bonewald LF (2013) The osteocyte: an endocrine cell… and more. Endocr Rev 34: 658-690. doi: 10.1210/er.2012-1026
    [77] Rochefort GY, Pallu S, Benhamou CL (2010) Osteocyte: the unrecognized side of bone tissue. Osteoporosis Int 21: 1457-1469. doi: 10.1007/s00198-010-1194-5
    [78] Rowe PSN (2012) Regulation of bone–renal mineral and energy metabolism: The PHEX, FGF23, DMP1, MEPE ASARM pathway. Crit Rev Eukaryot Gene Expr 22: 61-86. doi: 10.1615/CritRevEukarGeneExpr.v22.i1.50
    [79] Pajevic PD, Krause DS (2019) Osteocyte regulation of bone and blood. Bone 119: 13-18. doi: 10.1016/j.bone.2018.02.012
    [80] Frost HM (1987) The mechanostat: a proposed pathogenic mechanism of osteoporoses and the bone mass effects of mechanical and nonmechanical agents. Bone Miner 2: 73-85.
    [81] Tate MLK, Adamson JR, Tami AE, et al. (2004) The osteocyte. Int J Biochem Cell Biol 36: 1-8. doi: 10.1016/S1357-2725(03)00241-3
    [82] Bonewald LF, Johnson ML (2008) Osteocytes, mechanosensing and Wnt signaling. Bone 42: 606-615. doi: 10.1016/j.bone.2007.12.224
    [83] Manolagas SC, Parfitt AM (2010) What old means to bone. Trends Endocrinol Metab 21: 369-374. doi: 10.1016/j.tem.2010.01.010
    [84] Wang Y, Qin QH (2010) Parametric study of control mechanism of cortical bone remodeling under mechanical stimulus. Acta Mech Sinica 26: 37-44. doi: 10.1007/s10409-009-0313-z
    [85] Qu C, Qin QH, Kang Y (2006) A hypothetical mechanism of bone remodeling and modeling under electromagnetic loads. Biomaterials 27: 4050-4057. doi: 10.1016/j.biomaterials.2006.03.015
    [86] Parfitt AM (2002) Targeted and nontargeted bone remodeling: relationship to basic multicellular unit origination and progression. Bone 1: 5-7. doi: 10.1016/S8756-3282(01)00642-1
    [87] Hadjidakis DJ, Androulakis II (2006) Bone remodeling. Ann NYAcad Sci 1092: 385-396. doi: 10.1196/annals.1365.035
    [88] Vaananen HK, Zhao H, Mulari M, et al. (2000) The cell biology of osteoclast function. J cell Sci 113: 377-381.
    [89] Goldring SR (2015) The osteocyte: key player in regulating bone turnover. RMD Open 1: e000049. doi: 10.1136/rmdopen-2015-000049
    [90] Silver IA, Murrills RJ, Etherington DJ (1988) Microelectrode studies on the acid microenvironment beneath adherent macrophages and osteoclasts. Exp Cell Res 175: 266-276. doi: 10.1016/0014-4827(88)90191-7
    [91] Delaissé JM, Andersen TL, Engsig MT, et al. (2003) Matrix metalloproteinases (MMP) and cathepsin K contribute differently to osteoclastic activities. Microsc Res Techniq 61: 504-513. doi: 10.1002/jemt.10374
    [92] Delaisse JM (2014) The reversal phase of the bone-remodeling cycle: cellular prerequisites for coupling resorption and formation. Bonekey Rep 3: 561. doi: 10.1038/bonekey.2014.56
    [93] Bonewald LF, Mundy GR (1990) Role of transforming growth factor-beta in bone remodeling. Clin Orthop Relat R 250: 261-276.
    [94] Locklin RM, Oreffo ROC, Triffitt JT (1999) Effects of TGFβ and bFGF on the differentiation of human bone marrow stromal fibroblasts. Cell Biol Int 23: 185-194. doi: 10.1006/cbir.1998.0338
    [95] Lee B, Oh Y, Jo S, et al. (2019) A dual role of TGF-β in human osteoclast differentiation mediated by Smad1 versus Smad3 signaling. Immunol Lett 206: 33-40. doi: 10.1016/j.imlet.2018.12.003
    [96] Koseki T, Gao Y, Okahashi N, et al. (2002) Role of TGF-β family in osteoclastogenesis induced by RANKL. Cell Signal 14: 31-36. doi: 10.1016/S0898-6568(01)00221-2
    [97] Anderson HC (2003) Matrix vesicles and calcification. Curr Rheumatol Rep 5: 222-226. doi: 10.1007/s11926-003-0071-z
    [98] Bellido T, Plotkin LI, Bruzzaniti A (2019) Bone cells. Basic and Applied Bone Biology, 2 Eds Elsevier, 37-55. doi: 10.1016/B978-0-12-813259-3.00003-8
    [99] Weinstein RS, Jilka RL, Parfitt AM, et al. (1998) Inhibition of osteoblastogenesis and promotion of apoptosis of osteoblasts and osteocytes by glucocorticoids. Potential mechanisms of their deleterious effects on bone. J Clin Invest 102: 274-282. doi: 10.1172/JCI2799
    [100] Vezeridis PS, Semeins CM, Chen Q, et al. (2006) Osteocytes subjected to pulsating fluid flow regulate osteoblast proliferation and differentiation. Biochem Bioph Res Co 348: 1082-1088. doi: 10.1016/j.bbrc.2006.07.146
    [101] Lind M, Deleuran B, Thestrup-Pedersen K, et al. (1995) Chemotaxis of human osteoblasts: Effects of osteotropic growth factors. Apmis 103: 140-146. doi: 10.1111/j.1699-0463.1995.tb01089.x
    [102] Russo CR, Lauretani F, Seeman E, et al. (2006) Structural adaptations to bone loss in aging men and women. Bone 38: 112-118. doi: 10.1016/j.bone.2005.07.025
    [103] Ozcivici E, Luu YK, Adler B, et al. (2010) Mechanical signals as anabolic agents in bone. Nat Rev Rheumatol 6: 50-59. doi: 10.1038/nrrheum.2009.239
    [104] Rosa N, Simoes R, Magalhães FD, et al. (2015) From mechanical stimulus to bone formation: a review. Med Eng Phys 37: 719-728. doi: 10.1016/j.medengphy.2015.05.015
    [105] Noble BS, Peet N, Stevens HY, et al. (2003) Mechanical loading: biphasic osteocyte survival and targeting of osteoclasts for bone destruction in rat cortical bone. Am J Physiol-Cell Ph 284: C934-C943. doi: 10.1152/ajpcell.00234.2002
    [106] Robling AG, Castillo AB, Turner CH (2006) Biomechanical and molecular regulation of bone remodeling. Annu Rev Biomed Eng 8: 455-498. doi: 10.1146/annurev.bioeng.8.061505.095721
    [107] Qin QH, Mai YW (1999) A closed crack tip model for interface cracks inthermopiezoelectric materials. Int J Solids Struct 36: 2463-2479. doi: 10.1016/S0020-7683(98)00115-2
    [108] Yu SW, Qin QH (1996) Damage analysis of thermopiezoelectric properties: Part I—crack tip singularities. Theor Appl Fract Mec 25: 263-277. doi: 10.1016/S0167-8442(96)00026-2
    [109] Qin QH, Mai YW, Yu SW (1998) Effective moduli for thermopiezoelectric materials with microcracks. Int J Fracture 91: 359-371. doi: 10.1023/A:1007423508650
    [110] Jirousek J, Qin QH (1996) Application of hybrid-Trefftz element approach to transient heat conduction analysis. Comput Struct 58: 195-201. doi: 10.1016/0045-7949(95)00115-W
    [111] Qin QH (1995) Hybrid-Trefftz finite element method for Reissner plates on an elastic foundation. Comput Method Appl M 122: 379-392. doi: 10.1016/0045-7825(94)00730-B
    [112] Qin QH (1994) Hybrid Trefftz finite-element approach for plate bending on an elastic foundation. Appl Math Model 18: 334-339. doi: 10.1016/0307-904X(94)90357-3
    [113] Qin QH (2013)  Mechanics of Cellular Bone Remodeling: Coupled Thermal, Electrical, and Mechanical Field Effects CRC Press. doi: 10.1201/b13728
    [114] Wang H, Qin QH (2010) FE approach with Green's function as internal trial function for simulating bioheat transfer in the human eye. Arch Mech 62: 493-510.
    [115] Qin QH (2003) Fracture analysis of cracked thermopiezoelectric materials by BEM. Electronic J Boundary Elem 1: 283-301.
    [116] Qin QH, Ye JQ (2004) Thermoelectroelastic solutions for internal bone remodeling under axial and transverse loads. Int J Solids Struct 41: 2447-2460. doi: 10.1016/j.ijsolstr.2003.12.026
    [117] Qin QH, Qu C, Ye J (2005) Thermoelectroelastic solutions for surface bone remodeling under axial and transverse loads. Biomaterials 26: 6798-6810. doi: 10.1016/j.biomaterials.2005.03.042
    [118] Ducher G, Jaffré C, Arlettaz A, et al. (2005) Effects of long-term tennis playing on the muscle-bone relationship in the dominant and nondominant forearms. Can J Appl Physiol 30: 3-17. doi: 10.1139/h05-101
    [119] Robling AG, Hinant FM, Burr DB, et al. (2002) Improved bone structure and strength after long-term mechanical loading is greatest if loading is separated into short bouts. J Bone Miner Res 17: 1545-1554. doi: 10.1359/jbmr.2002.17.8.1545
    [120] Rubin J, Rubin C, Jacobs CR (2006) Molecular pathways mediating mechanical signaling in bone. Gene 367: 1-16. doi: 10.1016/j.gene.2005.10.028
    [121] Tatsumi S, Ishii K, Amizuka N, et al. (2007) Targeted ablation of osteocytes induces osteoporosis with defective mechanotransduction. Cell Metab 5: 464-475. doi: 10.1016/j.cmet.2007.05.001
    [122] Robling AG, Turner CH (2009) Mechanical signaling for bone modeling and remodeling. Crit Rev Eukar Gene 19: 319-338. doi: 10.1615/CritRevEukarGeneExpr.v19.i4.50
    [123] Galli C, Passeri G, Macaluso GM (2010) Osteocytes and WNT: the mechanical control of bone formation. J Dent Res 89: 331-343. doi: 10.1177/0022034510363963
    [124] Robling AG, Duijvelaar KM, Geevers JV, et al. (2001) Modulation of appositional and longitudinal bone growth in the rat ulna by applied static and dynamic force. Bone 29: 105-113. doi: 10.1016/S8756-3282(01)00488-4
    [125] Burr DB, Milgrom C, Fyhrie D, et al. (1996) In vivo measurement of human tibial strains during vigorous activity. Bone 18: 405-410. doi: 10.1016/8756-3282(96)00028-2
    [126] Sun W, Chi S, Li Y, et al. (2019) The mechanosensitive Piezo1 channel is required for bone formation. Elife 8: e47454. doi: 10.7554/eLife.47454
    [127] Goda I, Ganghoffer JF, Czarnecki S, et al. (2019) Topology optimization of bone using cubic material design and evolutionary methods based on internal remodeling. Mech Res Commun 95: 52-60. doi: 10.1016/j.mechrescom.2018.12.003
    [128] Goda I, Ganghoffer JF (2018) Modeling of anisotropic remodeling of trabecular bone coupled to fracture. Arch Appl Mech 88: 2101-2121. doi: 10.1007/s00419-018-1438-y
    [129] Louna Z, Goda I, Ganghoffer JF, et al. (2017) Formulation of an effective growth response of trabecular bone based on micromechanical analyses at the trabecular level. Arch Appl Mech 87: 457-477. doi: 10.1007/s00419-016-1204-y
    [130] Goda I, Ganghoffer JF (2017) Construction of the effective plastic yield surfaces of vertebral trabecular bone under twisting and bending moments stresses using a 3D microstructural model. ZAMM Z Angew Math Mech 97: 254-272. doi: 10.1002/zamm.201600141
    [131] Qin QH, Wang YN (2012) A mathematical model of cortical bone remodeling at cellular level under mechanical stimulus. Acta Mech Sinica-Prc 28: 1678-1692. doi: 10.1007/s10409-012-0154-z
  • This article has been cited by:

    1. Rehab Alharbi, S. E. Abbas, E. El-Sanowsy, H. M. Khiamy, K. A. Aldwoah, Ismail Ibedou, New soft rough approximations via ideals and its applications, 2024, 9, 2473-6988, 9884, 10.3934/math.2024484
    2. Ahmed Ramadan, Anwar Fawakhreh, Enas Elkordy, Novel categorical relations between $ \mathcal{L} $-fuzzy co-topologies and $ \mathcal{L} $-fuzzy ideals, 2024, 9, 2473-6988, 20572, 10.3934/math.2024999
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(11669) PDF downloads(1268) Cited by(18)

Figures and Tables

Figures(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog