Research article

On C-ideals and the basis of an ordered semigroup

  • Received: 26 February 2020 Accepted: 17 April 2020 Published: 21 April 2020
  • MSC : 06F05, 20M12

  • In this paper, we characterize ordered semigroups containing the greatest ideal and give the conditions of the greatest ideal being a C-ideal in an ordered semigroup. Moreover, we introduce the concept of a basis of an ordered semigroup and study the relationship between the greatest C-ideal and the basis in an ordered semigroup.

    Citation: Ze Gu, Xiang-Yun Xie, Jian Tang. On C-ideals and the basis of an ordered semigroup[J]. AIMS Mathematics, 2020, 5(4): 3783-3790. doi: 10.3934/math.2020245

    Related Papers:

    [1] Ahmet S. Cevik, Eylem G. Karpuz, Hamed H. Alsulami, Esra K. Cetinalp . A Gröbner-Shirshov basis over a special type of braid monoids. AIMS Mathematics, 2020, 5(5): 4357-4370. doi: 10.3934/math.2020278
    [2] Faiz Muhammad Khan, Weiwei Zhang, Hidayat Ullah Khan . Double-framed soft h-semisimple hemirings. AIMS Mathematics, 2020, 5(6): 6817-6840. doi: 10.3934/math.2020438
    [3] Fengxia Gao, Jialei Chen . Conjugacy classes of left ideals of Sweedler's four-dimensional algebra $ H_{4} $. AIMS Mathematics, 2022, 7(5): 7720-7727. doi: 10.3934/math.2022433
    [4] Aykut Or . Double sequences with ideal convergence in fuzzy metric spaces. AIMS Mathematics, 2023, 8(11): 28090-28104. doi: 10.3934/math.20231437
    [5] Ashraf S. Nawar, Mostafa A. El-Gayar, Mostafa K. El-Bably, Rodyna A. Hosny . θβ-ideal approximation spaces and their applications. AIMS Mathematics, 2022, 7(2): 2479-2497. doi: 10.3934/math.2022139
    [6] Tianyu Ni . On the gap between prime ideals. AIMS Mathematics, 2019, 4(1): 79-85. doi: 10.3934/Math.2019.1.79
    [7] Shahida Bashir, Ahmad N. Al-Kenani, Maria Arif, Rabia Mazhar . A new method to evaluate regular ternary semigroups in multi-polar fuzzy environment. AIMS Mathematics, 2022, 7(7): 12241-12263. doi: 10.3934/math.2022680
    [8] Faiz Muhammad Khan, Tian-Chuan Sun, Asghar Khan, Muhammad Junaid, Anwarud Din . Intersectional soft gamma ideals of ordered gamma semigroups. AIMS Mathematics, 2021, 6(7): 7367-7385. doi: 10.3934/math.2021432
    [9] Kai Wang, Guicang Zhang . Curve construction based on quartic Bernstein-like basis. AIMS Mathematics, 2020, 5(5): 5344-5363. doi: 10.3934/math.2020343
    [10] Jovanny Ibarguen, Daniel S. Moran, Carlos E. Valencia, Rafael H. Villarreal . The signature of a monomial ideal. AIMS Mathematics, 2024, 9(10): 27955-27978. doi: 10.3934/math.20241357
  • In this paper, we characterize ordered semigroups containing the greatest ideal and give the conditions of the greatest ideal being a C-ideal in an ordered semigroup. Moreover, we introduce the concept of a basis of an ordered semigroup and study the relationship between the greatest C-ideal and the basis in an ordered semigroup.


    Ideal theory play an important role in studying ordered semigroups. The concepts of ideals, bi-ideals, quasi-ideals, weakly prime ideals and prime ideals in ordered semigroups have been introduced by N. Kehayopulu in [1,2,3]. Moreover, fuzzy ideals, fuzzy bi-ideals, fuzzy quasi-ideals, weakly prime fuzzy ideals and prime fuzzy ideals in ordered semigroups have been studied in [4,5,6,7,8,9,10].

    The concept of C-ideals in semigroups has been introduced by I. Fabrici in [11]. The second author Xie of this paper extended the concept to ordered semigroups in [12]. Xie has studied some basic properties of C-ideals in ordered semigroups in [13] and characterized the ordered semigroups in which every proper ideal is a C-ideal in [12]. Mao, Xu and Lian have further studied some properties of C-ideals in ordered semigroups in [14]. Motivated by the previous work, in this paper, we study ordered semigroups containing the greatest ideal and give the conditions of the greatest ideal to be a C-ideal in an ordered semigroup. Moreover, we introduce the concept of a basis of an ordered semigroup and study the relationship between the greatest C-ideal and the basis in an ordered semigroup. If the order relation is trivial in an ordered semigroup S, then S is a semigroup. Consequently, all the results in this paper are true for semigroups.

    We call (S,,) an ordered semigroup if (S,) is a semigroup, (S,) is an ordered set and it satisfies:

    abaxbx,xaxb(a,b,xS).([15])

    We will just use S to denote an ordered semigroup when the operation and order are understood. A nonempty subset I of an ordered semigroup S is called an ideal if I satisfies: 1) IS,SII; 2) aS,abIaI. An ideal I of S is called proper if IS. A proper ideal M of S is called the greatest ideal if every proper ideal is contained in M. A proper ideal M of S is called a maximal ideal if whenever there exists an ideal N of S such that MN, then N=S. An ordered semigroup S is called simple if S contains no proper ideals. Let H be a nonempty subset of S. Denote

    (H]:={xShH,xh};[H):={xShH,hx}.

    If H has only one element a, then we denote (H] and [H) by (a] and [a) respectively. For any xS, denote by I(x) the ideal generated by x. Then we have I(x)=(xSxxSSxS] (see [1,2]).

    Green's relation J:={(x,y)S×SI(x)=I(y)} on an ordered semigroup S was introduced by N. Kehayopulu in [16]. It is easy to see that the relation J is an equivalence on S. For any xS, denote by Ix the J-class containing x. We define a relation "" on the set of all J-classes in S as below:

    IxIyI(x)I(y)(x,yS).

    It is routine to verify that "" is an order relation. A J-class Ix of S is called maximal if there is no other J-class Iy such that IxIy. A J-class Ix of S is called the greatest J-class if other J-classes are all contained in Ix. From [12], we know that M is a maximal ideal of S if and only if SM is a maximal J-class.

    Let S be an ordered semigroup. A proper ideal M of S is called a C-ideal if M(S(SM)S]. We know from [14] the following basic properties of C-ideals.

    Lemma 1. (1) (Theorem 5, [14]) If S is not simple, then S contains at least one C-ideal;

    (2) (Theorem 1, [14]) If S contains two different proper ideals M1,M2 such that M1M2=S, then neither of them is a C-ideal of S;

    (3) (Corollary 1, [14]) If S contains more than one maximal ideal, then none of them is a C-ideal of S;

    (4) (Theorem 4, [14]) If S contains only one maximal ideal M and M is a C-ideal, then M is the greatest ideal of S;

    (5) (Theorem 2, Theorem 3 [14]) If M1 and M2 are two C-ideals of S, then M1M2 and M1M2 are C-ideals of S.

    (6) (Theorem 7, [14]) If S contains the greatest ideal M and M is a C-ideal, then every proper ideal of S is a C-ideal.

    However, the following result given in [14] is incorrect, we next improve it.

    Result 1. (Theorem 6, [14]) Let S be an ordered semigroup with an identity 1. If S contains the greatest ideal, denoted by M, then M is a C-ideal or there exists aS(S3] such that S[a)M.

    Remark 1. If S has an identity 1, then S=(S3], i.e. S(S3]=. Consequently, we next improve Result 1 by the following result.

    Theorem 1. Let S be an ordered semigroup with an identity 1. Then every proper ideal of S is a C-ideal. In particular, if S contains the greatest ideal, denoted by M, then M is a C-ideal.

    Proof. Let M be a proper ideal of S. Then 1M. Suppose that 1M. Then S=S1SMM, i.e. S=M, this is a contradiction. Thus 1SM. It follows that (S(SM)S]=S. Therefore M(S(SM)S], i.e. M is a C-ideal.

    Since M is a proper ideal of S, we have M is a C-ideal.

    In the following, we study ordered semigroups containing the greatest ideal and give the conditions of the greatest ideal to be a C-ideal in an ordered semigroup.

    Lemma 2. If S(S2], then I(x)=(x] for any xS(S2].

    Proof. Let xS(S2]. Suppose that I(x)(x]. Then there exists aS or bS or c,dS such that ax>x or xb>x or cxd>x. Since ax,xb,cxdS2, we have x(S2]. This is a contradiction.

    Theorem 2. If S(S2], then S contains the greatest ideal if and only if S=(a] for some aS(S2].

    Proof. () Denote by M the greatest ideal of S. Let aSM. Since (S2]M, aS(S2]. Moreover, Ia=SM is the greatest J-class. Thus I(x)I(a) for any xM and so MI(a). In addition, I(a)M=S. It follows that S=I(a)=(a] from Lemma 2.

    () Let S=(a] for some aS(S2]. Next we prove that S{a} is an ideal of S. In fact: for any x,yS and bS{a}, we have xb,byS{a}. Otherwise, a=xb=byS2 which contradicts that aS(S2]. Let cS{a} and zS. If zc, then zS{a}. Otherwise, z=a which contradicts that c<a. It follows that S{a} is an ideal of S.

    Finally, we show that S{a} is the greatest. Let I be a proper ideal of S. Then aI and thus IS{a}.

    Theorem 3. Let M be the greatest ideal of S.

    (1) If S=(S2], then M is a C-ideal.

    (2) If S(S2], then M=S{a} for some aS(S2].

    Proof. (1) Since (S(SM)S] is an ideal of S and M is the greatest ideal of S, we have (S(SM)S]=S or (S(SM)S]M. We distinguish three cases.

    1) If (S(SM)S]=S, then M(S(SM)S]. Hence M is a C-ideal.

    2) If (S(SM)S]=M, then M is a C-ideal clearly.

    3) If (S(SM)S]M, then (S3]=(S(SM)SSMS](S(SM)S](SMS]M(M]=MM=MS. Since S=(S2], we have S=(S3] which contradicts (S3]S.

    (2) It can be easily obtained from the proof of Theorem 2.

    Corollary 1. Let M be the greatest ideal of S. If M is a C-ideal, then (S2]=(S3]. In particular, if S(S2], then M is a C-ideal if and only if M=(S3].

    Proof. If S=(S2], then (S2]=(S3] obviously. Next we consider the case of S(S2]. From Theorem 3, we know that M=S{a} for some aS(S2]. Since M is a C-ideal and (S2] is a proper ideal, we have (S2]M(S(SM)S]=(SaS](S3](S2]. It follows that M=(S2]=(S3].

    Let S(S2]. We need only prove the sufficiency. We know that S=(a] for some aS(S2] from Theorem 2 and M=S{a} from Theorem 3(2). Since M=(S3], we have M=(S3]=(S(a]S]=(SaS]=(S(SM)S]. Thus M is a C-ideal.

    In this section, we introduce the concept of a basis of an ordered semigroup and study the relation between the existence of the greatest C-ideal and the existence of a basis in an ordered semigroup.

    Definition 1. A nonempty subset A of an ordered semigroup S is called a basis of S if A satisfies the following conditions:

    1) (ASAASSAS]=S;

    2) There is no proper subset B of A such that (BSBBSSBS]=S.

    Example 1. We consider the order semigroup S={a,b,c,d} with the multiplication "" and the order relation "" below:

    ≤:={(a,a),(a,b),(a,c),(a,d),(b,b),(c,c),(d,d)}.

    Let A={c,d}. Then (ASAASSAS]=S. However, d(cSccSScS] and c(dSddSSdS]. Thus A is a basis of S.

    Theorem 4. Let A be a nonempty subset of S. Then A is a basis of S if and only if A satisfies:

    1) For any xS, there exists aA such that I(x)I(a);

    2) For any a1,a2A, if Ia1Ia2, then a1=a2.

    Proof. () Obviously, condition 1) can be obtained from Definition 1. Next we prove condition 2).

    If Ia1Ia2, then I(a1)I(a2). Suppose that a1a2. Let B=A{a1}. Then (ASAASSAS](BSBBSSBS]. Therefore (BSBBSSBS]=S. This contradicts Definition 1.

    () By the condition 1), we have

    S=xSI(x)aAI(a)=(ASAASSAS]S.

    Hence (ASAASSAS]=S. Suppose that there exists BA such that (BSBBSSBS]=S. Let a1AB. Then there exists b1B or b2B or b3B or b4,b5B,sS such that a1b1 or a1sb2 or a1b3s or a1b4sb5. It implies that I(a1)I(b1) or I(a1)I(b2) or I(a1)I(b3) or I(a1)I(b4), i.e. Ia1Ib1 or Ia1Ib2 or Ia1Ib3 or Ia1Ib4. This contradicts 2).

    Remark 2. By Theorem 4, we can see that if A is a basis of S, then every element of A belongs to some maximal J-class and there is only one element in A for every maximal J-class.

    Proposition 1. Let {MααΛ} be a set of all maximal ideals of S, ˆM=αΛMα and ¯Mα=SMα. Then we have

    1) S=(αΛ¯Mα)ˆM;

    2) If αβ, then ¯Mα¯Mβ=;

    3) If αβ, then ¯MαMβ;

    4) If αβ, then ¯Mα¯MβˆM, i.e. ˆM;

    5) Let I be an ideal of S and I¯Mα, then ¯MαI.

    Proof. 1) Since

    ˆM=αΛMα=αΛ(S¯Mα)=SαΛ¯Mα,

    we have

    S=(αΛ¯Mα)ˆM.

    2) We know that ¯Mα is a maximal J-class for any αΛ. Thus ¯Mα¯Mβ= when αβ.

    3) It can be obtained from 2) obviously.

    4) Let αβ,μα¯Mα,μβ¯Mβ and μ=μαμβˆM. By 1), we have S=(αΛ¯Mα)ˆM. Thus there exists μγ¯Mγ such that μ=μγ.

    i) If γα, then ¯MαMγ. Thus ¯Mα¯MβMγ. Hence μγMγ which contradicts μγ¯Mγ.

    ii) If γ=α, then ¯MβMγ. It follows that ¯Mα¯MβMγ Thus μγMγ which is also a contradiction.

    By i) and ii), we get μ=μαμβˆM. Hence ¯Mα¯MβˆM, i.e. ˆM.

    5) If I is an ideal of S and I¯Mα, then MαI=S. Thus, ¯MαI.

    Remark 3. 1) If S contains the greatest C-ideal, denote it by Mg, then MgˆM. Indeed: If there exists αΛ such that MgMα, then MgMα=S. By Lemma 1(2), we can see that Mg is not a C-ideal. This is a contradiction.

    2) The greatest C-ideal Mg does not necessarily exist in all ordered semigroups. See the following example.

    Example 2. Let S={a,b,c,d,e} with the multiplication "" and the order "≤" below:

    ≤:={(a,a),(a,b),(b,b),(c,c),(d,b),(d,c),(d,d),(e,c),(e,e)}.

    It is easy to check that (S,,) is an ordered semigroup and {a},{d},{e} are C-ideals of S. However, there is not the greatest C-ideal.

    In the following, we give the relationship between the basis and the greatest C-ideal Mg of S.

    Theorem 5. Let S contain a basis A. Then S contains the greatest C-ideal Mg. Moreover, Mg=(S3]ˆM where ˆM=αΛMα and {MααΛ} is the family of all maximal ideals of S.

    Proof. For any aA, Ia is a maximal J-class. Thus SIa is a maximal ideal. Hence ˆM. Since ˆM and (S3] are ideals of S, ˆM(S3]. Denote N=ˆM(S3]. Then for any xN, there exists cS such that x(ScS]. If cA, then there exists bA such that I(c)I(b). Therefore c(bSbbSSbS]. Obviously, cb. Next we distinguish two cases.

    1) If c<b, then (ScS](SbS]. Thus x(SbS].

    2) If c(SbbSSbS], then (ScS](S(SbbSSbS]S](S2bSSbS2S2bS2](SbS]. Hence x(SbS].

    By 1) and 2), we have x(SbS](SAS](S(SˆM)S](S(SN)S] (Because SˆM=SαΛMα=αΛ¯Mα and every element of A is in some maximal J-class). Consequently, N is a C-ideal.

    Finally, we prove that N is the greatest. Let T be a C-ideal of S. Then T(S(ST)S](S3]. By Lemma 1(4), we have every C-ideal is contained in all maximal ideals. Thus TˆM. Then T(S3]ˆM. It follows that N=(S3]ˆM is the greatest C-ideal Mg.

    Proposition 2. Let S contain the greatest C-ideal Mg and Mg(S3]. Then every J-class of (S3]Mg is maximal and I(a)=(SaS] for any a(S3]Mg.

    Proof. Since (S3] and Mg are ideals of S, (S3]Mg is a union of some J-classes of S. Let Mγ be any one of them. Then Mγ(S3]. It implies that there exist x,y,bS such that axby for any aMγ. Thus a(SbS]. Next we show bMγ. If there exists δγ such that bMδ, then a(SbS] which implies that I(a)I(b) and bI(a). Otherwise, bI(a). Hence I(b)I(a) and so I(a)=I(b) which contradicts δγ. Therefore bSI(a). It follows that a(S(SI(a))S]. Then I(a)(S(SI(a))S], i.e. I(a) is a C-ideal. Therefore I(a)Mg is a C-ideal properly containing Mg, which is impossible. Thus bMγ. This implies I(a)(SbS]I(b)=I(a), hence I(a)=(SbS]=I(b). Next we show that I(a)=(SaS].

    1) If a=b, then I(a)=(SaS].

    2) If ab, then b(aSaaSSaS].

    i) If b<a, then I(b)=(SbS](SaS]I(a)=I(b). Thus I(a)=(SaS].

    ii) If b(SaaSSaS], then I(b)=(SbS](S(SaaSSaS]S](S2aSSaS2S2aS2](SaS]I(a)=I(b). Hence I(a)=(SaS].

    Finally, we show that Mγ is maximal. Suppose that I(a)=(SaS]I(c) for aMγ(S3]Mg and some cS. Then aI(c)=(cSccSScS]. Obviously, ac. We show that I(a)(ScS].

    1) If a<c, then I(a)=(SaS](ScS].

    2) If a(SccSScS], then I(a)=(SaS](S(SccSScS]S](S2cSScS2S2cS2](ScS].

    Since cI(a), we have I(a)(S(SI(a))S], i.e. I(a) is a C-ideal. Therefore I(a)Mg is a C-ideal properly containing Mg, this is a contradiction. It follows that Mγ is a maximal J-class.

    Theorem 6. Let S contain the greatest C-ideal Mg. If S(S2] and any two elements of S(S2] are incomparable, then S contains a basis.

    Proof. Firstly, we have Mg(S(SMg)S](S3](S2]S. We denote by Mα a J-class of S(S2], by Mβ a J-class of (S2](S3] and by Mγ a J-class of (S3]Mg.

    1) From Lemma 2, we have

    I(x)=I(y)(x]=(y]x=y(x,yS(S2]).

    Thus there is only one element in Mα. Since any two elements of S(S2] are incomparable, Mα is maximal.

    2) For any xMβ(S2](S3], there exist u,vS such that xuv. Here u,vS(S2]. Otherwise u,v(S2], then x(S3] which is a contradiction. Therefore xI(u)=(u].

    3) From Proposition 2, we get that Mγ is maximal. Since Mg(S(SMg)S], there exists zSMg such that yI(z) for any yMg. It follows that yMα or Mβ or Mγ. Next we construct a set A.

    i) If zMα or zMγ, then we choose z into A.

    ii) If zMβ, then there exists uMα, such that zI(u). Hence we can choose u into A.

    Now, we have

    Mg(ASAASSAS];(S2](S3](ASAASSAS];S(S2](ASAASSAS];(S3]Mg(ASAASSAS].

    Therefore S(ASAASSAS], i.e. (ASAASSAS]=S.

    To prove that A is a basis of S, it remains to show that there is no proper subset BA with the property (BSBBSSBS]=S. This is evident, because A is constructed by means of elements of maximal J-classes of S, and from each maximal J-class just one element is chosen into A.

    Ideal theory play an important role in studying ordered semigroups. In this paper, we first study ordered semigroups containing the greatest ideal and give the conditions of the greatest ideal to be a C-ideal in an ordered semigroup. Furthermore, we introduce the concept of a basis of an ordered semigroup and establish the relationship between the greatest C-ideal and the basis in an ordered semigroup.

    We thank the referees whose comments led to significant improvements to this paper and the editor for his/her warm work.

    This research was supported by the National Natural Science Foundation of China (No. 11701504, 11801081); the Natural Science Foundation of Guangdong Province (No. 2014A030313625, No. 2018A030313063); the Project for Characteristic Innovation of 2018 Guangdong University (No. 2018KTSCX234); the Teaching Team Project in Guangdong Province (Guangdong teach highletter [2018] no. 179); the Demonstration Project of Grass-roots Teaching and Research Section in Anhui Province (No. 2018jyssf053); the University Natural Science Project of Anhui Province (No. KJ2018A0329) and the Innovation Education Project of Graduate Students of Guangdong Province (No. 2016SFKS-40).

    The authors declare no conflict of interest.



    [1] N. Kehayopulu, On weakly prime ideals in ordered semigroups, Math. Japon., 35 (1990), 1050-1056.
    [2] N. Kehayopulu, On prime, weakly prime ideals in ordered semigroups, Semigroup Forum, 44 (1992), 341-346. doi: 10.1007/BF02574353
    [3] N. Kehayopulu, On completely regular poe-semigroups, Math. Japon., 37 (1992), 123-130.
    [4] N. Kehayopulu, M. Tsingelis, Fuzzy sets in ordered groupoids, Semigroup Forum, 65 (2002), 128-132. doi: 10.1007/s002330010079
    [5] N. Kehayopulu, M. Tsingelis, Fuzzy bi-ideals in ordered semigroups, Inform. Sci., 171 (2005), 13-28. doi: 10.1016/j.ins.2004.03.015
    [6] N. Kehayopulu, M. Tsingelis, Regular ordered semigroups in terms of fuzzy subsets, Inform. Sci., 176 (2006), 3675-3693. doi: 10.1016/j.ins.2006.02.004
    [7] X. Y. Xie, J. Tang, Fuzzy radicals and prime fuzzy ideals of ordered semigroups, Inform. Sci., 178 (2008), 4357-4374. doi: 10.1016/j.ins.2008.07.006
    [8] X. Y. Xie, J. Tang, Regular ordered semigroups and intra-regular ordered semigroups in tems of fuzzy subsets, Iranian J. Fuzzy Syst., 7 (2010), 121-140.
    [9] M. Shabir, A. Khan, Characterizations of ordered semigroups by the properties of their fuzzy ideals, Comput. Math. Appl., 59 (2010), 539-549. doi: 10.1016/j.camwa.2009.06.014
    [10] M. Shabir, A. Khan, Fuzzy quasi-ideals of ordered semigroups, Bull. Malays. Math. Sci. Soc, 34 (2011), 87-102.
    [11] I. Fabrici, Semigroups containing covered two-sided ideals, Math. Slovaca, 34 (1984), 355-363.
    [12] M. F. Wu, X. Y. Xie, On C-ideals of ordered semigroups, J. Wuyi University (in Chinese), 9 (1995), 43-46.
    [13] X. Y. Xie, M. F. Wu, On po-semigroups containing no maximal ideals, Southeast Asian Bull Math., 20 (1996), 31-36.
    [14] H. Y. Mao, X. Z. Xu, X. P. Lian, On C-ideals of ordered semigroups, J. Shandong University (in Chinese), 45 (2010), 14-16.
    [15] L. Fuchs, Partially Ordered Algebraic Systems, New York: Pregamon Press, 1963.
    [16] N. Kehayopulu, Note on Green's relations in ordered semigroups, Math. Japon., 36 (1991), 211-214.
  • 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(4316) PDF downloads(319) Cited by(0)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog