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Volume 14, Issue 1 (Jan 2016)

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Fuzzy ideals of ordered semigroups with fuzzy orderings

Xiaokun Huang
  • College of Mathematics and Econometrics, Hunan University, Changsha 410082, China and College of Mathematics, Honghe University, Mengzi 661199, China
  • Email:
/ Qingguo Li
  • Corresponding author
  • College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
  • Email:
Published Online: 2016-11-14 | DOI: https://doi.org/10.1515/math-2016-0076

Abstract

The purpose of this paper is to introduce the notions of ∈, ∈ ∨qk-fuzzy ideals of a fuzzy ordered semigroup with the ordering being a fuzzy relation. Several characterizations of ∈, ∈ ∨qk-fuzzy left (resp. right) ideals and ∈, ∈ ∨qk-fuzzy interior ideals are derived. The lattice structures of all ∈, ∈ ∨qk-fuzzy (interior) ideals on such fuzzy ordered semigroup are studied and some methods are given to construct an ∈, ∈ ∨qk-fuzzy (interior) ideals from an arbitrary fuzzy subset. Finally, the characterizations of generalized semisimple fuzzy ordered semigroups in terms of ∈, ∈ ∨qk-fuzzy ideals (resp. ∈, ∈ ∨qk-fuzzy interior ideals) are developed.

KeyWords: Fuzzy set, (∈, ∈ ∨qk)-Fuzzy ideal; (∈, ∈ ∨qk)-Fuzzy interior ideal; Generalized semisimple fuzzy ordered semigroup

MSC 2010: 03E72; 06F05; 08A72

1 Introduction

Based on Zadeh’s fuzzy set theory, the study of fuzzy algebraic structures has started in the pioneering paper of Rosenfeld [1] in 1971. Rosenfeld [1] introduced the notion of fuzzy groups and showed that many results in groups can be extended in an elementary manner to develop the theory of fuzzy group. Since then the literature of various fuzzy algebraic concepts has been growing very rapidly. Using the notion “belongingness (∈)” and “quasi-coincidence (q)” of a fuzzy point with a fuzzy set proposed by Pu and Liu [2], the concept of (α, β)-fuzzy subgroups, where α, β are any two of {∈,q, ∈ ∨q, ∈ ∧q} with α ≠∈∧q, was introduced by Bhakat and Das [3] in 1992. Particularly, it was pointed out in the same paper that (∈, ∈ ∨q)-fuzzy subgroup is an important and useful generalization of Rosenfeld’s fuzzy subgroup. After that, these generalizations have been extended to other algebraic structures by many researchers, for example, Davvaz [4], Jun and Song [5], Kazancl and Yamak [6], Khan and Shabir [7], Yin et al. [8], Zhan and Yin [9], Davvaz and Khan [10], etc. As a generalization of the quasi-coincident relation (q) of a fuzzy point with a fuzzy subset, Jun [11] defined (∈, ∈ ∨qk)-fuzzy subalgebras in BCK/BCI-algebras. In [12] (∈, ∈ ∨qk)-fuzzy h-ideals and (∈, ∈ ∨qk)-fuzzy k-ideals of a hemiring are defined and discussed. Shabir et al. [13] characterized different classes of semigroups by the properties of their (∈, ∈ ∨qk)-fuzzy ideals and (∈, ∈ ∨qk)-fuzzy bi-ideals.

By an ordered semigroup we mean a semigroup together with a partial order that is compatible with the semigroup operation. Since ordered semigroup has a close relation with theoretical computer science, especially with the theory of sequential machines, formal languages, computer arithmetics, and error-correcting codes, it has been extensively investigated by many researchers (see e.g. [1422]). On the other hand, the concept of fuzzy orderings was introduced and investigated at the very beginning by Zadeh [23]. The motivation for Zadeh [23] to do this research was that the crisp partial orders are useful tools for modeling various situations in which different kinds of comparison appear, while fuzzy partial orders are more sensitive than their crisp counterparts. During the past 10 years, many authors investigated fuzzy orders with their applications to various branches of mathematics and computer sciences. Some examples are as follows. Zhang et al. [24, 25] analysed the properties of fuzzy domains and fuzzy complete lattices. Lai [26], Yao [27] and Hao [28] investigated the relationships between fuzzy orders and fuzzy topological spaces. Based on fuzzy posets, Yao et al. [29, 30] studied the fuzzy Scott topologies and fuzzy Galois connections. Since ordered semigroups and fuzzy orderings both have applications across a wide variety of fields, researchers turn to consider the combination of these two concepts. In [31], Hao defined fuzzy ordered semigroups based on fuzzy partial orders and studied their representation by sets. After that, Wang [32] solved the problem of embedding such a fuzzy ordered semigroup into a fuzzy quantales.

In this paper, the method of (∈, ∈ ∨qk)-fuzzy generalization is applied to fuzzy ordered semigroups in which the orderings are fuzzy relations. We introduce the concepts of (∈, ∈ ∨qk)-fuzzy ideals and (∈, ∈ ∨qk)-fuzzy interior ideals of a fuzzy ordered semigroup and present several characteristic theorems of them. We describe the (∈, ∈ ∨qk)-fuzzy ideals and (∈, ∈ ∨qk)-fuzzy interior ideals generated by a fuzzy subset and the structures of all (∈, ∈ ∨qk)-fuzzy (interior) ideals of a fuzzy ordered semigroup. In order to exhibit some applications of (∈, ∈ ∨qk)-fuzzy (interior) ideals, we investigate the characterizations of generalized semisimple fuzzy ordered semigroup by the properties of (∈, ∈ ∨qk)-fuzzy (interior) ideals.

2 Basic definitions and results

Recall that an ordered semigroup is a partially ordered set S, with an associative multiplication “.” which is compatible with the ordering, i.e., for any x,y,zS,xy implies xzyz and zxzy.

For two subsets A, B of S, we denote (A]={tS:th,for somehA} and AB={ab:aA,bB}. A nonempty subset A of S is called a left (resp. right) ideal of S if: 1SAA(resp.ASA)and(2)baA implies bA. An ideal of S is a non-empty subset which is both a left and right ideal of S. A nonempty subset A of an ordered semigroup S is called an interior ideal of S if: (1)SASA,(2)AAAand(3)baA implies bA.

Let X be a nonempty set. An arbitrary mapping f:X[0,1] is called a fuzzy subset of X and the symbol F(X) denotes the set of all fuzzy subsets of X. For α(0,1]andAX, the fuzzy subset αA is defined by αA(x)=αifxAandαA(x)=0 otherwise. In particular, when α=1,αA is said to be the characteristic function of A, denoted by fA;whenA={x},αA is said to be afuzzy point with support x and value α and denoted by xα.

A fuzzy point xα. is said to belong to (resp. be quasi-coincident with) a fuzzy subset f, denoted as xαf(resp.xαqf),iff(x)α(resp.f(x)+α>1). In [11], Jun generalized the concept of xαqfasxαqkfiff(x)+α+k>1, where k[0,1).Ifxαforxαqkf, then we write xαqkf. By the symbol xα¯f we mean that xαf does not hold, where {,qk,qk}.

Now, using the above notions, we define an ordering “⊆ ∨qk” on F(X). Let f,gF(X). If for any fuzzy point xαonX,xαfimpliesxαqkg, then we write fqkg.

Lemma 2.1: Let f,gF(X).fqkgifandonlyifg(x)f(x)1k2forallxX.

Proof: Let fqkg and x be an element of X. If possible, let g(x)<f(x)1k2. Then there exists a real number α such that g(x)<α<f(x)1k2 which yields that xαfbutxαqk¯g, a contradiction. Thus g(x)f(x)1k2forallxX.Conversely, assume that g(x)f(x)1k2forallxX.Letxα be any fuzzy point such that xαf. Then g(x)f(x)1k2α1k2.Ifα1k2,theng(x)α,i.e.,xαg.Ifα1k2,theng(x)1k2, which implies g(x)+α>1k2+1k2=1k, and hence xαqkg. Thus, in both cases we always have xαqkg. Therefore, fqkg.The following lemma can be obtained directly from Lemma 2.1.

Lemma 2.2: Let f,g,hF(X). Then(1) fqkf.(2) fqkgandgqkhimplyfqkh.It is natural to ask wether fqkgandgqkfimplyf=g. The following example gives a negative answer to this question.

Example 2.3: Let f and g be twofuzzy subsets of a set X=a,b,csuchthatfa=0.6,fb=0.6,fc=0.5andga=0.5,gb=0.5,gc=0.6. Then, for any k[0,1),wehavefqkgandgqkf,butfg.Let f, g ∈ F(X). We define a relation on F(X) as follows. If fqkgandgqkf,then we writefkg. It follows from Lemma 2.2 that ≡ k is an equivalence relation on F(X).

Definition 2.4: ([24, 34 Assume that X is a nonempty set. A fuzzy relation e : X×X[0,1] on X is called an fuzzy partial order iffor any x, y, z ∈ X,(1) (Reflexivity) e(x, x) = 1;(2) (Transitivity) e(x,y)e(y,z)e(x,z);(3) (Anti-symmetry) e(x,y)=e(y,x)=1impliesx=y.A nonempty set X equipped with a fuzzy partial order is called afuzzy partially ordered set, or shortly afuzzy poset.

Definition 2.5: Definition 2.5 ([31, 32]). A fuzzy ordered semigroup is a triple (S, e ⋅) consisting of a nonempty set S together with a fuzzy relation e and a binary operationson S such that(1) (S, e) is a fuzzy poset;(2) (S, ·) is a semigroups;(3) e(x,y)e(zx,zy)ande(x,y)e(xz,yz)forallx,y,zS.A fuzzy ordered semigroup (S, e, ⋅) is said to be commutative if x ⋅ y = y ⋅ x for all x, y ∈ S. By the identity of S we mean an element I ∈ S such that x ⋅ I = I ⋅ x = x for all x ∈ S. In what follows, for the sake of simplicity, we shall write xy instead of x ⋅ y, for any x, y ∈ S.

Example 2.6: Let S be a nonempty set, ⩽ be a binary relation on S and ⋅ be a binary operation on S. For α[0,1), we define a fuzzy relation e⩽ on S as follows: (x,y)S×S, e(x,y)=1,xyα,otherwise then (S, e ⩽,⋅) is a fuzzy ordered semigroup if and only if (S, e ⩽,⋅) is a crisp ordered semigroup.In the sequel, the symbol θ denotes the Godel implication on [0, 1], i.e., θ(r,t)=1,rtt,r>t,r,t0,1 (see [33]). Then, a binary operation on F(S), which is induced directly by θ, can be defined by θ(f,g)(x)=θ(f(x),g(x))foranyf,gF(S)andxS.

Example 2.7: Define a fuzzy relation e : [0,1]×[0,1][0,1]by(x,y[0,1],e(x,y)=θ(x,y)) and a binary operation . : [0,1]×[0,1][0,1]by(x,y[0,1],xy=xy). Then ([0, 1], e, ⋅) is a commutative fuzzy ordered semigroup with the identity 1.

Definition 2.8: ([31]). Let (S, e·) be a fuzzy ordered semigroup and fF(S). Then f is called a fuzzy left (resp. right) ideal of S if the following conditions hold: (1) f(y)f(x)e(y,x)foranyx,yS; (2) f(xy)f(y)(resp.f(xy)f(x))foranyx,yS.By a fuzzy ideal, we mean the one which is both a fuzzy left and fuzzy right ideal.Let (S,e,୵) be a fuzzy ordered semigroup. We define a unary operation and a binary operation (·] on F(S), respectively, as follows: f,gF(S),xS,(1) (f](x)=yXe(x,y)f(y).(2) (feg)(x)=(y,z)S×Sf(y)e(x,yz)g(z)Particularly, for an element x of S, we shall write (x] instead of f{x}]. It is clear that (x](y)=zXe(y,z)f{x}(z)=e(y,x)In addition, if ( f] = f, then f is called afuzzy lower set (See [24, 34]).

Remark 2.9: Let (S,e,୵) be a fuzzy ordered semigroup. Then(1) ∘e is associative on F(S), that is, (foeg)oeh=foe(goeh) for any f,g,h∈F(S).(2) If S is a commutative fuzzy ordered semigroup, thene is commutative on F(S), that is, foeg=goef for any f,g, ∈ F(S).Finally, we develop some basic properties of the multiplication ∘e. We will omit the proofs because they are trivial.

Proposition 2.10: Let (S,e,୵) be a fuzzy ordered semigroup and f,g,hiF(S), iI. Then(1) (foeg]=foeg.(2) foe(ihi)=ifoehi,(ihi)oef=ihioef.(3) foe(ihi)ifoehi,(ihi)oefihioef.The item (1) in above proposition indicates that for any f;gF(S),foeg is a fuzzy lower set, and the item (2) shows that ∘>e is distributive over arbitrary unions.

Proposition 2.11: Let (S,e,୵) be a fuzzy ordered semigroup and fi;giF(S)(i=1,2)such thatf1qkf2andg1qkg2.(1) f1g1qkf2g2.(2) f1g1qkf2g2.(3) f1oeg1qkf2oeg2.

Proposition 2.12: Let (S,e,୵) be a fuzzy ordered semigroup and A,BS. Then(1) ABifandonlyiffAqkfB. (2) fAfB=fABandfAfB=fAB.(3) fAefB=(fAB].

3 (∈, ∈ ∨ qk)-fuzzy left (resp. right) ideals

To avoid repetitions, from now $S$ will always mean a fuzzy ordered semigroup (S,e,୵)

Definition 3.1: A fuzzy subset $f$ of a fuzzy ordered semigroup S is called an (∈, ∈ ∨ qk)-fuzzy left (resp. right) ideal of S iffor any x, y ∈ S and α ∈ (0,1)], the following conditions hold.(F1a) xαfyαqkθ(x],f.(F2a) ySandxαfyxαqkfresp.xyαqkf.A fuzzy subset $f$ of $S$ is called an (∈, ∈ ∨qk)-fuzzy ideal if it is both an (∈, ∈ ∨qk)-fuzzy left ideal and (∈, ∈ ∨ qk)-fuzzy right ideal of $S$. We note that, whenever a statement is made about (∈, ∈ ∨qk)-fuzzy left ideals, analogous statement holds for (∈, ∈ ∨qk)-fuzzy right ideals.In what follows, we develop some characterizations of (∈, ∈ ∨qk)-fuzzy left ideals of a fuzzy ordered semigroup.

Proposition 3.2: Let S be a fuzzy ordered semigroup and fF(S). Then f is an (∈, ∈ ∨qk-fuzzy left ideal of S if and only if(Flb) f(y)f(x)e(y,x)1k2foranyx,yS;(F2b) f(xy)f(y)1k2foranyx,yS.

Proof: (⟹) Let f be an (∈, ∈ ∨qk-fuzzy left ideal of S. If f(y)<f(x)e(y,x)1k2for somex,yS, then there exists α[0,1]such thatf(y)<α<f(x)e(y,x)1k2. Then it follows that xαf. Meanwhile, since f(y)<e(y,x),wehaveθ((x],f)(y)=θ(e(y,x),f(y))=f(y). This yields the following two assertions.(i) yα¯θ((x],f).Indeed,sinceθ((x],f)(y)=f(y)<α,wehaveyα¯θ((x]cf).(ii) yαqk¯θ((x],f). In fact, a routine calculation implies that θ((x],f)(y)+α+k=f(y)+α+k<1k2+1k2+k=1, thus we have yαqk¯θ((x],f).Hence, yαqk¯θ((x],f), which contradicts to (Fla). Therefore f(y)f(x)e(y,x)1k2 for any x,yS. In a similar way, we can prove that f(x,y)f(y)1k2for anyx,yS.(⟸) Suppose that (Flb) and (F2b) hold. Let x,yS x,yS and α ∈ (0,1] be such that xαf. Then f(x)α. By condition (Flb), we have f(y)αe(y,x)1k2.Ifαe(y,x)>1k2,thenf(y)1k2, which implies θ((x],f)(y)+α+k=θ(e(y,x),f(y))+α+kf(y)+α+k>1,and thenyαqkθ((x],f). If αe(y,x)1k2,thenf(y)f(x)e(y,x),it derivesf(y)f(x)orf(y)e(y,x).Sincef(y)f(x) implies θ((x],f)(y)=θ(e(y,x),f(y))f(y)f(x)αandf(y)e(y,x)impliesθ((x],f)(y)=1α, we always have yαθ((x],f).Henceyαqkθ((x],f). This means that (Fla) holds. Similarly, we can prove that (F2a) holds.Note that every fuzzy left (resp. right) ideal of S according to Definition 2.8 is an (∈, ∈ ∨qk-fuzzy left (resp. right) ideal of S. However, the following example reveals that an (∈, ∈ ∨qk-fuzzy left (resp. right) ideal is not necessarily a fuzzy left (resp. right) ideal.

Example 3.3: Consider the fuzzy ordered semigroup S = {a,b,c}, where fuzzy order e and multiplication · are defined respectively as follows:

Let f be a fuzzy subsets of S such thatf=0.6a+0.8b+0.2c Put $k=0.4$. Then it is easy to veri that f is an (∈, ∈ ∨qk-fuzzy left ideal of S, but not a fuzzy left ideal of S because f(ab) = f(a) = 0.6 < 0.8 = f(b).

Proposition 3.4: Let (S,e,·) be a fuzzy ordered semigroup and fF(S) . Then f is an (∈, ∈ ∨qk)-fuzzy left ideal of S if and only iffor each α[0,1k2], fα is either empty or a left ideal of the ordered semigroup (S,α;) where fα={xS:f(x)α}andα={(x,y)S×S:e(x,y)α}.

Proof: (⟹) Let α[0,1k2] Then it is easy to check that (S,α;) is an ordered semigroup. Now, if fα ≠ ⊘, we show that fα is aleft ideal of (S,α;). Let x, yS be such that yαxandxfα,thene(y,x)αandf(x)α. Since f is an (∈, ∈ ∨qk-fuzzy left ideal, it follows from Proposition 3.2 that f(y)f(x)e(y,x)1k2α, which implies yfα.Next, letxSandyfα. Then f(xy)f(y)1k2α is a left ideal of (S,α;).(⟸) For x,yS, set α=f(x)e(y,x)1k2,thenα[0,1k2] and fα is nonempty. By the assumption, fα is a left ideal of (S,α;). Since yαxandxfα,We have yf_α, which implies that f(y)α=f(x)e(y,x)1k2. Similarly, we have f(xy)f(y)1k2by settingα=f(y)1k2. Therefore, by Proposition 3.2, f is an (∈, ∈ ∨qk-fuzzy left ideal of (S, e, ·).

Proposition 3.5: Let S be a fuzzy ordered semigroup and fF(S) . Then f is an (∈, ∈ ∨qk)-fuzzy left ideal of S if and only if(F1C)(f]qkf;(F2(C)) fSefqkf.

Proof: (⟹) Let f be an (∈, ∈ ∨qk-fuzzy left ideal of S and xS. Then, by Proposition 3.2, we have (f](x)1k2=ySe(x,y)f(y)1k2f(x)and(fSef)(x)1k2=(y,Z)S×Sex,yzf(z)1k2 (y,z)S×Sex,yzf(yz)1k2fx. It follows from Lemma 2.1 that (f]qkf; and fSefqkf, i.e., (Flc) and (F2c) hold.(⟸) Assume that (Flc) and (F2c) hold. If possible, let f(y)<f(x)e(y,x)1k2 for some x,yS. Then there exists a real number α such that f(y)<α<f(x)e(y,x)1k2, which implies yα(f],yα¯f and yαqk¯f, a contradiction to (Flc). Therefore, the condition (Flb) holds. Similarly, we can prove that (F2b) is valid.Let (S,e,#183;) be a fuzzy ordered semigroup and xS. Recall from Section 2 that the fuzzy subset (x] is defined by (x](y)=e(y,x) for all yS. Now we generalize it as follows. For any xS and α ∈ (0,1], U(x;α) is a fuzzy subset of S defined by U(x;α)(y)=αe(y,x) for all yS. One may easily observe that U(x;1) = (x] and xαU(x;α).

Proposition 3.6: Let S be a fuzzy ordered semigroup and fF(S) . Then f is an (∈, ∈ ∨qk)-fuzzy left ideal of S if and only iffor any x,yS and α ∈ (0,1], the following conditions hold.(F1d)xαfU(x;α)qkf;(F2d) yS,xαfU(yx;α)qkf.

Proof: (⟸) Let f be an (∈, ∈ ∨qk-fuzzy left ideal of S and let x, yS and α∈(0,1] be such that xαf. Then f(x) ⩾ α. By Proposition 3.2, we have f(y)f(x)e(y,x)1k2αe(y,x)1k2=U(x;α)(y)1k2. Thus Lemma 2.1 implies that U(x;α)qkf, and hence (Fld) holds. Moreover, for any zS, we have f(z)f(yx)e(z,yx)1k2f(x)e(z,yx)1k2αe(z,yx)1k2=U(yx;α)(z)1k2. Thus U(yx;α)qkf. So (F2d)$ holds.(⟸) Assume that (F1d) and (F2d) hold. Let x,yS and set α=f(x)e(y,x)1k2 Then xαf. By the assumption, we have U(x;α)qkf, which implies f(y)U(x;α)(y)1k2=αe(y,x)1k2=f(x)e(y,x)1k2 Next, we set α=f(x)1k2 Then (F2d) implies U(yx;α)qkf. Thus we have f(yx)U(yx;α)(yx)1k2=αe(yx,yx)1k2=α1k2=f(x)1k2. Therefore, by Proposition 3.2, f is an (∈, ∈ ∨qk-fuzzy left ideal of S.In the following, we intend to construct an (∈, ∈ ∨qk-fuzzy left ideal from an arbtrary fuzzy subset of a fuzzy ordered semigroup.

Proposition 3.7: Let S be a fuzzy ordered semigroup and fF(S). Then (1k2)sef is an (∈, ∈ ∨qk-fuzzy left ideal of S.Straightforward by Propositions 2.10 and 3.5.

Theorem 3.8: Let S be a fuzzy ordered semigroup and fF(S) . Put f(x)=α[0,1]|(α1k2;x[fα]α)or(α>1k2;xfα) for any xS. Thenfis the least (∈, ∈ ∨qk-fuzzy left ideal of S that contains f. Where, for any subset A of S, [A]α is the left ideal of the ordered semigroup (S,α;) generated by A, i.e., [A]α={xS|xαyforsomeyASA}.

Proof: We will complete the proof by three steps.Step 1: 〈f〉 is an (∈, ∈ ∨qk-fuzzy left ideal of S. For any x,yS, define two sets A1={αe(y,x)(1k2)|α[0,1],(α1k2;x[fα]α)or(α>1k20xfα)};A2={β|β[0,1],(β1k2;y[fβ]β)or(β>1k2;yfβ)}.Let γ=αe(y,x)(1k2)A1 Then α1k2;x[fα]αorα>1k2;xfα.Ifα1k2,x[fα]α, then γ[0,1k2],andγαyieldsx[fα]α[fγ]γ. If α>1k2;xfα,thenγ[0,1k2]andxfαfγ[fγ]γ. So, in both cases, we always have γ[0,1k2]andx[fγ]γ. Since e(y,x)γ, we have y[fγ]γ Thus γA2. This proves that A1A2. Thus, f(y)A2A1=f(x)e(y,x)1k2. In a similar way, we can prove that f(xy)f(y)1k2 for any x,yS. Therefore, 〈f〉 is an (∈, ∈ ∨qk-fuzzy left ideal of S.Step 2: f{f}. For any xS, since fα[fα]α for every α ∈[0,1], we have f(x)=α[0,1]|xfα={α[0,1]|(α1k2,xfα)or(α>1k2,xfα)}{α[0,1]|(α1k2,x[fα]α)or(α>1k2,xfα)}=f(x).This implies f ⊆ 〈f〉, as required.Step 3: 〈f〉 ⊆ g for any (∈,∈ ∨ qk-fuzzy left ideal g of S with fg. For any α[0,1k2] it follows from Proposition 3.4 that gα is either empty or a left ideal of the ordered semigroup (S,⩽α In both cases, we always have gα = [gα]α. Thus, for any xS, we have f(x)={α[0,1]|(α1k2,x[fα]α)or(α>1k2,xfα)}{α[0,1]|(α1k2,x[gα]α)or(α>1k2,xgα)}={α[0,1]|(α1k2,xgα)or(α>1k2,xgα)}={α[0,1]|xgα}=g(x).1This implies that 〈f〉 ⊆ g.Naturally, we can consider the greatest (∈, ∈ ∨qk-fuzzy left ideal of a fuzzy ordered semigroup contained in a fuzzy set. For this, we have the following result.

Theorem 3.9: Let f be any fuzzy subset of S. Then i(f)(x)={α[0,1]|U(x;α)qkf,U(ax;α)qkf(aS)} is the greatest (∈, ∈ ∨qk-fuzzy left ideal of S contained (under the relation ⊆ ∨ qk in f.

Proof: For x,yS, define B1={αe(y,x)(1k2)|α[0,1],U(x;α)qkf,U(ax;α)qkf(aSB2={α[0,1]|U(y;α)qkf,U(by;α)qkf(bSB3={α(1k2)|α[0,1],U(x;α)qkf,U(cx;α)qkf(cSNow, we split the proof into three parts as follows.(i) i(f) is an (∈, ∈ ∨qk-fuzzy left ideal of S. For any γ=αe(y,x)(1k2)B1, we have U(x;α)qkf,U(ax;α)qkf(aS). Thus, for any zS,f(z)U(x;α)(z)(1k2)=αe(z,x)(1k2)αe(z,y)e(y,x)(1k2)=U(y;αe(y,x)(1k2))(z)(1k2)=U(y;γ)(z)(1k2)andf(z) Uax;αz1k2=αez,ax1k2αez,ayeay,ax1k2αez,ayey,x(1k2)=U(ay,αe(y,x)(1k2)(z)(1k2)=U(ay,γ)(z)(1k2),aS, which implies Uy;γqkf,Uay,yqkf,aS. Thus γB2. This proves that B1B2 and hence i(f)(y)=B2B1=i(f)(x)e(y,x)1k2. In a similar way, we can prove i(f)(xy)i(f)(y)1k2. Therefore, i(f) is an (∈, ∈ ∨qk-fuzzy left ideal of S.(ii) i(f)qkf For any γ=α(1k2)B3, we have U(x;α)qkf. Combining Lemma 2.1, we get that f(x)U(x;α)(x)(1k2)=α(1k2)=γ. Since γ is an arbitrary element of B3, we have f(x)B3=i(f)(x)(1k2), i.e.,i(f)qkf.(iii) gi(f) for any (∈, ∈ ∨qk-fuzzy left ideal g that is contained (under ⊆ ∨qk) in f. For any fuzzy point xαg, since g is an (∈, ∈ ∨qk)-fuzzy left ideal, it follows from Proposition 3.6 that U(x;α)qkg and U(ax;α)qkgfor anyaS. Thus, for any xS, we have g(x)={α[0,1]|g(x)α}={α[0,1]|xαg}{α[0,1]|U(x;α)qkg,U(ax;α)qkg(aS)}{α[0,1]|U(x;α)qkf,U(ax;α)qkf(aS)}=i(f)(x).6This implies gi(f), as required.The following two propositions are easy to prove.

Proposition 3.10: Let {fi : iI be a family of (∈, ∈ ∨qk)-fuzzy left ideals of a fuzzy ordered semigroup S. Then ifiandifi are both (∈, ∈ ∨qk)-fuzzy left ideals of S.

Proposition 3.11: Let f and g be two (∈, ∈ ∨qk)-fuzzy left ideals of afuzzy ordered semigroup S. Then so is foeg.Let Fidl (S) be the set of all (∈, ∈ ∨qk-fuzzy ideals of S. Then it follows from Theorem 3.8, 3.9 and Proposition 3.10 that f={g|gFidl(S),fg}and(f)={g|gFidl(S),gqkf}.Let us recall that a quantale is a triple (Q,*,⩽) such that (Q, ⩽) is a complete lattice, (Q,*) is a semigroup and for any xQ and {yi}i∈IQ, x(iyi)=ixyiand(iyi)x=iyix.Combing Proposition 2.10, 3.10 and 3.11, we obtain the following theorem.

Theorem 3.12: Let S be a fuzzy ordered semigroup. Then (Fidl (S), ⊆, ∘e) is a quantale.

4 (∈, ∈ ∨qk)-fuzzy interior ideals

Definition 4.1: Let S be a fuzzy ordered semigroup and fF(S) . Then f is called an (∈, ∈ ∨qk-fuzzy interior ideal of S if it satisfies (Fla) and for any x,y,zS and α, β ∈ (0,1], the following conditions hold. (F3a)xα;YBf(xy)αBqkf;(F4a)yαf(xyz)αqkf.

Proposition 4.2: Let S be a fuzzy ordered semigroup and fF(S) . Then f is an (∈, ∈ ∨qk-fuzzy interior ideal of S if and only if (F1b)f(y)f(x)e(y,x)1k2foranyx,yS;(F3b)f(xy)f(x)f(y)1k2foranyx,yS;(F4b)f(xyz)f(y)1k2foranyx,y,zS.

Proof: The proof is similar to that of Proposition 3.2. □

Proposition 4.3: Let (S, e, ⋅) be a fuzzy ordered semigroup and f ∈ F(S). Then f is an (∈, ∈ ∨qk)-fuzzy interior ideal of S if and only if for each α[0,1k2],fα is either empty or an interior ideal of the ordered semigroup S,α,.

Proof: The proof runs parallel to that of Proposition 3.4. □

Proposition 4.4: Let S be a fuzzy ordered semigroup and f ∈ F(S). Then f is an (∈, ∈ ∨qk-fuzzy interior ideal of S if and only if (F1c) (f]qkf; (F3c) fefqkf; (F4c) fSoefoefSqkf.

Proof: The proof is similar to that of Proposition 3.5. □

Proposition 4.5: Let S be a fuzzy ordered semigroup and f ∈ F(S). Then f is an (∈, ∈ ∨qk)-fuzzy interior ideal of S if and only if for any x, y, z ∈ S and α ∈ (0, 1], the following conditions hold. (F1d) xαfU(x;α)qkf; (F3d) xα,yBfU(xy;αβ)qkf; (F4d) yαfU(xyz;α)qkf.

Proof: The proof runs parallel to that of Proposition 3.6. □

Theorem 4.6: Let S be a fuzzy ordered semigroup and f ∈ F(S). Put f(x)={α[0,1]|(α1k2;x[fα]α)or(α>1k2;xfα)} for any x ∈ S. Then 〈f〉 is the least (∈, ∈ ∨qk)-fuzzy interior ideal of S that contains f, where, for any subset A of S, [A]α is the intersection of all crisp interior ideals of the ordered semigroup (S, ⩽α, ⋅) containing A.

Proof: The proof is similar to that of Theorem 3.8. □

Proposition 4.7: Let {fi:iI} be a family of (∈, ∈ ∨qk)-fuzzy interior ideals of a fuzzy ordered semigroup S. Then ifi is an (∈, ∈ ∨qk)-fuzzy interior ideal of S.

Proof: It is easy to verify. □The following example indicates that ifi is not an (∈, ∈ ∨qk)-fuzzy interior ideal of S, in general.

Example 4.8: Consider the fuzzy ordered semigroup S={a,b,c,d}, in which fuzzy order e and multiplication . are defined respectively as follows:

Let f and g be two fuzzy subsets of S such that f=0.3a+0.3b+0c+0d,g=0.3a+0b+0.3c+0dThen, for any k ∈ [0, 1), both f and g are (∈, ∈ ∨qk)-fuzzy interior ideals of S, but f ⋃ g is not an (∈, ∈ ∨qk)-fuzzy interior ideal of S, since (fg)(bc)=(fg)(d)=f(d)g(d)=0<0.31k2=(fg)(b)(fg)(c)1k2.In what follows, we discuss the structures of all (∈, ∈ ∨qk)-fuzzy interior ideals of a fuzzy ordered semigroup. By the way, we will give an answer to the question which raises naturally from Example 4.8, that is, under which conditions ifi is an (∈, ∈ ∨qk)-fuzzy interior ideal?Before doing this, we first review some basic concepts in lattice theory.A nonempty subset P of an ordered set L is said to be directed if for every pair of elements x, y ∈ P, there exists an element z ∈ P such that x ⩽ z and y ⩽ z.

Definition 4.9: ([35]). A non-empty family L of a set X is said to be a topped ⋂-structure if(1) XL.(2) iIAiLforanynonemptyfamily{Ai}iIofL.(3) iIAiLforanydirectedfamily{Ai}iIofL.As a fuzzy version of above definition, we present the following.

Definition 4.10: Let F be a nonempty family of fuzzy subsets of a set X. Then F is said to be a topped algebraic fuzzy-structure ⋂-structure if(1) XF.(2) iIfiFforanynonemptysubfamily{fi}iIofF.(3) iIfiFforanydirectedfamily{fi}iIofF.Let FIidl(S) be the set of all (∈, ∈ ∨qk)-fuzzy interior ideals of S and FIidl(S).

Theorem 4.11: Let (S,e,) be a fuzzy ordered semigroup. Then (FIidl(S),) is a topped algebraic fuzzy ⋂-structure.

Proof: Clearly, by Proposition 4.7, the items (1) and (2) in Definition 4.10 hold in (FIidl(S),). Now, let {fi}iI be a directed family of FIidl(S). We prove that iIfiFIidl(S). In fact, let f=iIfi, then we obtain: (i) Since each fi(iI) is an (∈, ∈ ∨qk)-fuzzy interior ideal of S, for any x, y ∈ S, we have fxey,x1k2=iIfixey,x1k2=iI(fixey,x1x2)iIfiy=iIfiy=fy. (ii) For any x, y ∈ Q, we have f(xy)=iIfixy=iIfi(xy)iI(fi(x)fi(y)1k2)=iIfixiIfiy1k2Δ=fxfy1k2.Now, we prove that Equation (¶) holds. It is clear that iIfixfiy1k2iIfixiIfiy1k2. If possible, let iIfixfiy1k2iIfixiIfiy1k2. Then iIfixfiy1k2<iIfixiIfiy1k2, and hence there exists a real number r such that iIfixfiy1k2<r<iIfixiIfiy1k2. On one hand, since r<iIfixiIfiy1k2, there exists i, j ∈ I such that r<fixfiy1k2. Since {fi}iI is directed, there exists kI such that fifk and fjfk Thus, r<fkxfky1k2. On the other hand, form iIfixfiy1k2<r, we have fixfiy1k2<r for every i ∈ I, a contradiction. So, iIfixfiy1k2=iIfixiIfiy1k2.(iii) For any x, y, z ∈ Q, we have fxyz=iIfixyz=iIfixyziI(fiy1k2)=(iIfi(y))1k2=fy1k2.Therefore, f=iIfi is an (∈, ∈ ⋅qk)-fuzzy interior ideal of S, i.e., iIfiFIidlS.

Corollary 4.12: Let {fi:iI} be a directed family of (∈, ∈ ⋅qk)-fuzzy interior ideals of a fuzzy ordered semigroup S. Then ifi is an (∈, ∈ ⋅qk)-fuzzy interior ideal of S.Let L be a complete lattice. An element x ∈ L is said to be compact, if for any subset PLwithyP, there exists a finite subset QPsuchthatxQ. A complete lattice L is said to be an algebraic lattice, if every x ∈ L can be written as ajoin of compact elements. The following result is well known in lattice theory.

Proposition 4.13: ([35]). Every topped algebraic ⋂-structure is an algebraic lattice.However, using the concept of (∈, ∈ ⋅qk)-fuzzy interior ideals, we can give an example to illustrate that a topped algebraic fuzzy ⋂-structure is generally not an algebraic lattice.

Example 4.14: Let S be a fuzzy ordered semigroup. By Theorem 4.11, (FIidl (S), ⊆) is a topped algebraic fuzzy ⋂-structure and it is trivial that (FIidl (S), ⊆) is a complete lattice. Now, let f ∈ FIidl (S) be such that f. Define a family offuzzy subsets as follows: gn(x)=(11n+1)f(x),xS,nN.Then it is easy to veri that gn is an (∈, ∈ ⋅qk)-fuzzy interior ideal of S for every nNand{gn}nN is a directed family. Thus, by Theorem 4.11, we have f=n=1gn=n=1gn. However, f can not be expressed as a join (equivalently a union) of any finite sub-family of {gn}nN. Thus f is not a compact element in FIidl (S). Therefore (FIidl (S), ⊆) is not an algebraic lattice because there is no compact element in it more than ∅.

Proposition 4.15: Let (S, ⩽, ⋅) be an ordered semigroup. Given any chain of subsets S0S1Sn=S and any chain of relations 01...n=of such that Sj is an interior ideal of ordered semigroup (S, ⩽, ⋅) for every j ∈ {1, ..., n}. Then there exist a fuzzy subset of S f and a fuzzy relation e such that f is an (∈, ∈ ⋅qk)-fuzzy interior ideal offuzzy ordered semigroup (S, ⩽, ⋅) in which the level interior ideals of f are exactly the chain S0S1Sn=Sandf1k2=S.

Proof: Let {αi[0,1k2]|,i=0,1,;n} be such that 1k2=α0>α1>>αn. Let f be a fuzzy subset of S and e be a fuzzy relation on S defined respectively by f(x)=α0,ifxS0α1,ifxS1S0αn,ifxSnSn1ande(x,y)=α0,if(x,y)∈⩽0α1,if(x,y)∈⩽10αn,if(x,y)nn1 for all x, y ∈ S. Then it is not difficult to verify that e is a fuzzy partial order on S such that (S, e, ⋅) is a fuzzy ordered semigroup, andfα=S0,ifα[α0,1]S0,ifα[α1,α0)Sn,ifα[α1,αn)andα=0,ifα[α0,1]0,ifα[α1,α0)n,ifα[0,αn)Hence, by the hypothesis, fα is an interior ideal of ordered semigroup (Sα #183;) for every α0,1k2. Thus Proposition 4.3 implies that f is an (∈,∈ ∨ qk)-fuzzy interior ideal of fuzzy ordered semigroup (S,e,·), and all level interior ideals of f are exactly the chain S0S1Sn=Sandf1k2=S.Concerning the relationship between (∈,∈ ∨ qk)-fuzzy ideals and (∈,∈ ∨ qk)-fuzzy interior ideals of a fuzzy ordered semigroup, we have the following result.

Proposition 4.16: Let (S,e,·) be a fuzzy ordered semigroup. Then each (∈,∈ ∨ qk)-fuzzy ideal of S is an (∈,∈ ∨ qk)-fuzzy interior ideal of S.

Proof: It is straightforward.The converse of Proposition 4.16 is not true in general as shown in the following example.

Example 4.17: Consider the fuzzy ordered semigroup S={a,b,c,d}, where fuzzy order e and multiplication · are defined respectively as follows:

Let f be a fuzzy subsets of S such that f=0.8a+0.3b+0.6c+0dThen f is an $(\in_{\mathrm{D}}\in\ \vee q_{k})$-fuzzy interior ideal but not an (∈,∈ ∨ qk)-fuzzy ideal of S for any $k\in[0,1$), because f(cd)=f(b)=0.3<0.6=0.61k2=f(c)1k2.It is a natural question: under which conditions does the converse of Proposition 4.2 hold? We will answer this question in the next section.

5 Application: characterizations of fuzzy ordered semigroups

As a generalization of semisimple semigroups, the concept of semisimple (crisp) ordered semigroups is proposed by Shabir and Khan in [20]. An ordered semigroup (S,⩽,·) is called semisimple if for any xS, there exist a,b,cS such that xaxbxc. Using fuzzy partial orders, we can now develop a more generalized type of semisimple ordered semigroups.

Definition 5.1: A fuzzy ordered semigroup (S,e,·) is said to be generalized semisimple, if for any xS, a,b,cse(x,axbxc)1k2.Let (S,⩽,·) be a semisimple ordered semigroup in the sense of Shabir and Khan, and e ⩽ be the characteristic function of ⩽. Then it is easy to check from Definition 5.1 that (S,e·) is a generalized semisimple fuzzy ordered semigroup. This means that semisimple ordered semigroup is a special case of generalized semisimple fuzzy ordered semigroup.More generally, we can characterize the relationship between a generalized semisimple fuzzy ordered semigroup and a semisimple ordered semigroup as follows.

Proposition 5.2: A fuzzy ordered semigroup (S,e,·) is generalized semisimple if and only iffor any α[0,1k2], (S,⩽α·) is a semisimple ordered semigroup.

Proof: The proof is straightforward by Definition 5.1.Next we answer the question that we proposed in the previous section.

Proposition 5.3: Let S be a generalized semisimple fuzzy ordered semigroup. Then every (∈,∈ ∨ qk)-fuzzy interior ideal is an $(\in,\ \in\vee q_{k})$-fuzzy ideal of S.

Proof: Let f be an (∈,∈ ∨ qk)-fuzzy interior ideal of S and x,yS. Then it follows from Propositions 3.2 and Definition 5.1 that f(xy)zsf(z)e(xy,z)1k2a,b,csf(axbxcy)e(xy,axbxcy)1k2a,b,csf(x)e(xy,axbxcy)1k2a,b,csf(x)e(x,axbxc)1k2=f(x)[a,b,cse(x,axbxc)]1k2f(x)1k2.Similarly, we can prove that f(xy)f(y)1k2. Thus f is an (∈,∈ ∨ qk)-fuzzy ideal of S.Combining Propositions 4.16 and 5.3, we can conclude that, in any generalized semisimple fuzzy ordered semigroups the concepts of (∈,∈ ∨ qk)-fuzzy ideals and (∈,∈ ∨ qk)-fuzzy interior ideals coincide with each other.

Proposition 5.4: A fuzzy ordered semigroup (S,e,·) is generalized semisimple if and only if fkfoef for every (∈,∈ ∨qk)-fuzzy ideal f of S.

Proof: (⟸) Suppose S is a fuzzy ordered semigroup. Let x be any element of S. Then it is easy to verify that g=:(f{x}S{x}{x}SS{x}S] is an (∈,∈ ∨ qk)-fuzzy ideal f of S. Thus, by assumption, we have gkgeg. It follows Proposition 2.12 that (f{x}S{x}{x}SS{x}S]qk(f{x}S{x}{x}SS{x}S]oe(f{x}S{x}{x}SS{x}S]=(f({x}S{x}{x}SS{x}S)({x}S{x}{x}SS{x}S)]=(f{x2}{x}S{x}{x2}S{x}S{x}SS{x2}S{x}S{x}S{x2}SS{x}S{x}S], where the last equality follows from the fact that ({x}S{x}{x}SS{x}S)({x}S{x}{x}SS{x}S)={x2}{x}S{x}{x2}S{x}S{x}SS{x2}S{x}S{x}S{x2}SS{x}S{x}S{x}S{x}{x}S2{x}{x}S{x}S{x}S2{x}SS{x}S{x}S{x}S2{x}S{x}S{x}SS{x}S2{x}S={x2}{x}S{x}{x2}S{x}S{x}SS{x2}S{x}S{x}S{x2}SS{x}S{x}S.Then, from Lemma 2.1 it follows that 1k2(f{x2}U{x}S{x}U{x2}SU{x}S{x}SUS{x2}US{x}S{x}US{x2}SUS{x}S{x}S](x)Further, by the item (2) of Proposition 2.12, we have 1k2(f{x2}](x)or1k2(f{x}S{x}](x)or 1k2(f{x2}S](x)or1k2(f{x}S{x}S](x)or1k2(fS{x2}](x)or1k2(fS{x}S{x}](x)or 1k2(fS{x2}S](x)or1k2(fS{x}S{x}S](x).If1k2(f{x2}](x),then1k2e(xWx2). Since e(x,x2)e(x,x2)e(x2,x3)e(x3,x4)e(x4,x5), we have 1k2e(x,x2)e(x,x5)a,b,cSe(x,axbxc).if1k2bSe(x,xbx)bSe(x,xbxbxbx)a,b,cSe(x,axbxc).In a similar way, we can prove that a,b,cSe(x,axbxc)1k2 holds for all cases we listed above.Consequently, S is a generalized semisimple fuzzy ordered semigroup.(⟹) Let f be an (∈,∈ ∨ qk)-fuzzy ideal of S. Then it follows from Proposition 3.5 that fefqkfSefqkf. On the other hand, for any xS, we have (fef)(x)=y,zS×Sfye(x,yz)f(z)a,b,cSfaxe(x,axbxc)f(bxc)a,b,cSf(x)e(x,axbxc)f(x)1k2,fx1k2, implying that fqkfef,and hencefkfef.

Theorem 5.5: A fuzzy ordered semigroup S is generalized semisimple if and only iffor any two (∈,∈ ∨ qk)-fuzzy interior ideals f and g of S, fgkfeg.

Proof: (⟹) Assume that S is a generalized semisimple fuzzy ordered semigroup. Let f and g be (∈,∈ ∨ qk)-fuzzy interior ideals of S. Then both f and g are (∈,∈ ∨ qk)-fuzzy ideals of S. By Proposition 3.5, we have fegqkfefSqkfandfegqkfSegqkg. This derives that fegqkfg. Now, let $x$ be an element of S. Then (feg)(x)=y,zSf(y)e(x,yz)g(z)a,b,cSf(axb)e(x,(axb)xc)g(xc)a,b,cSf(x)e(x,axbxcy)g(x)1k2f(x)g(x)1k2=(fg)g(x)1k2.This proves that fgqkfeg,and hencefgkfeg.(⟸) Assume that the given condition holds. Let f be any (∈,∈ ∨ qk)-fuzzy ideals of S. Then, by Proposition 4.16, f is also an (∈,∈ ∨ qk)-fuzzy interior ideals of S. The hypothesis implies that f=ffkfef. Therefore, S is generalized semisimple.Combining Propositions 5.3 and Theorem 5.5, we obtain the following theorem.

Theorem 5.6: S be a fuzzy ordered semigroup. Then the following conditions are equivalent.(1) S is generalized semisimple.(2) fgkfeg for any (∈,∈ ∨qk)-fuzzy ideals f and g of S.(3) fgkfeg for any (∈,∈ ∨qk)-fuzzy interior ideals f and g of S.(4) fgkfeg for every (∈,∈ ∨qk)-fuzzy ideal f and every (∈,∈ ∨qk)-fuzzy interior ideal g of S.(5) fgkfeg for every (∈,∈ ∨qk)-fuzzy interior ideal f and every (∈,∈ ∨qk)-fuzzy ideal g of S.6) The set of all (∈, ∈∨ qk-fuzzy ideals of S forms a semilattice under the multiplicatione and the relationk on F(S), that is, fegkgefandfkfef for all (∈,∈ ∨qk)-fuzzy ideals f and g of S.(7) The set of all (∈,∈ ∨ qk)-fuzzy interior ideals of S forms a semilattice under the multiplicatione and the relationk on F(S), that is, fegkgefandfkfef for all (∈,∈ ∨qk)-fuzzy interior ideals f and g of S.

Theorem 5.7: Let S be a fuzzy ordered semigroup. Then the following conditions are equivalent.(1) S is generalized semisimple.(2) fgqkfeg for every (∈,∈ ∨qk)-fuzzy left ideal f and every (∈,∈ ∨qk)-fuzzy interior ideal g of S.(3) fgqkfeg for every (∈,∈ ∨qk)-fuzzy left ideal f and every (∈,∈ ∨qk)-fuzzy ideal g of S.(4) fgqkfeg for every (∈,∈ ∨qk)-fuzzy interior f and every (∈,∈ ∨qk)-fuzzy right ideal g of S.(5) fgqkfeg for every (∈,∈ ∨qk)-fuzzy ideal f and every (∈,∈ ∨qk)-fuzzy right ideal g of S.

Proof: Assume that (1) holds. Let x be an element of S. Then (feg)(x)=y,zS×Sfye(x,yz)g(z)a,b,cSfaxe(x,ax(bxc))g(bxc)a,b,cSf(x)e(x,axbxcy)g(x)1k2,fxgx1k2=fggx1k2.This implies that fgqkfeg and hence (2) holds. In a similar way, we can show that (4) holds. And it is clear that (2) ⟹ (3) and (4) ⟹ (5).Now, assume that (3) holds. Let f be an (∈,∈ ∨qk-fuzzy ideal of S. Then f is also an (∈,∈ ∨qk-fuzzy left ideal of S, hence we have f=ffqkfef On the other hand, Proposition 4.2 implies fefqkf. Thus fkfef, that is (1) holds. In a similar way, we can prove that (5)⟹(1). This completes the proof.

Acknowledgement

This work is supported by the National Natural Science Foundation of China (No. 11371130) and Research Fund for the Doctoral Program of Higher Education of China (No. 20120161110017).

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About the article

Received: 2016-03-03

Accepted: 2016-09-13

Published Online: 2016-11-14

Published in Print: 2016-01-01



Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0076. Export Citation

© Huang and Li, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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