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.
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. [14–22]). 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
For two subsets A, B of S, we denote
Let X be a nonempty set. An arbitrary mapping
A fuzzy point
Now, using the above notions, we define an ordering “⊆ ∨qk” on F(X). Let f,g ∈ F(X). If for any fuzzy point
Let
Let
Conversely, assume that
The following lemma can be obtained directly from Lemma 2.1.
Let
(1)
(2)
It is natural to ask wether
Let f and g be twofuzzy subsets of a set
Let f, g ∈ F(X). We define a relation on F(X) as follows. If
([24, 34 Assume thatXis a nonempty set. A fuzzy relatione :
(1) (Reflexivity) e(x, x) = 1;
(2) (Transitivity)
(3) (Anti-symmetry)
A nonempty set X equipped with a fuzzy partial order is called afuzzy partially ordered set, or shortly afuzzy poset.
Definition 2.5 ([31, 32]). A fuzzy ordered semigroup is a triple (S, e ⋅)consisting of a nonempty setStogether witha fuzzy relationeand a binary operations ⋅ onSsuch that
(1) (S, e) is a fuzzy poset;
(2) (S, ·)is a semigroups;
(3)
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.
LetSbe a nonempty set, ⩽ bea binary relation onS and ⋅ be a binary operation on S. For
In the sequel, the symbol θ denotes the Godel implication on [0, 1], i.e.,
Define a fuzzy relatione :
([31]). Let(S, e·)be a fuzzy ordered semigroup and
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:
(1)
(2)
Particularly, for an element x of S, we shall write (x] instead of f{x}]. It is clear that
In addition, if ( f] = f, then f is called afuzzy lower set (See [24, 34]).
Let (S,e,୵) be a fuzzy ordered semigroup. Then
(1) ∘eis associative onF(S), that is,
(2) IfSis a commutative fuzzy ordered semigroup, then ∘eis commutative onF(S), that is,
Finally, we develop some basic properties of the multiplication ∘e. We will omit the proofs because they are trivial.
Let (S,e,୵) be a fuzzy ordered semigroup andf,g,hi ∈ F(S), i ∈ I. Then
(1)
(2)
(3)
The item (1) in above proposition indicates that for any
Let (S,e,୵) be a fuzzy ordered semigroup and
(1)
(2)
(3)
Let (S,e,୵) be a fuzzy ordered semigroup and
(1)
(2)
(3)
3 (∈, ∈ ∨ qk)-fuzzy left (resp. right) ideals
To avoid repetitions, from now $S$ will always mean a fuzzy ordered semigroup (S,e,୵)
A fuzzy subset $f$ of a fuzzy ordered semigroupSis called an (∈, ∈ ∨ qk)-fuzzy left (resp. right) idealofSiffor anyx, y ∈ Sand α ∈ (0,1)], the following conditions hold.
(F1a)
(F2a)
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.
LetSbe a fuzzy ordered semigroup andf ∈ F(S). Thenfis an (∈, ∈ ∨qk-fuzzy left ideal ofSif and only if
(Flb)
(F2b)
(⟹) Let f be an (∈, ∈ ∨qk-fuzzy left ideal of S. If
(i)
(ii)
Hence,
(⟸) Suppose that (Flb) and (F2b) hold. Let x,y ∈ Sx,y ∈ S and α ∈ (0,1] be such that xα ∈ f. Then
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.
Consider the fuzzy ordered semigroupS = {a,b,c}, where fuzzy ordereand multiplication · are defined respectively as follows:
Letfbe a fuzzy subsets ofSsuch that
Put $k=0.4$. Then it is easy to veri thatfis an (∈, ∈ ∨qk-fuzzy left ideal ofS, but not a fuzzy left ideal of S because f(ab) = f(a) = 0.6 < 0.8 = f(b).
Let(S,e,·)be a fuzzy ordered semigroup andf ∈ F(S) . Thenfis an (∈, ∈ ∨qk)-fuzzy left idealofSif and only iffor each
(⟹) Let
(⟸) For x,y ∈ S, set
LetSbe a fuzzy ordered semigroup andf ∈ F(S) . Thenfis an (∈, ∈ ∨qk)-fuzzy left ideal ofSif and only if
(F1C)
(F2(C))
(⟹) Let f be an (∈, ∈ ∨qk-fuzzy left ideal of S and x∈S. Then, by Proposition 3.2, we have
(⟸) Assume that (Flc) and (F2c) hold. If possible, let
Let (S,e,#183;) be a fuzzy ordered semigroup and x∈S. Recall from Section 2 that the fuzzy subset (x] is defined by
LetSbe a fuzzy ordered semigroup andf ∈ F(S) . Thenfis an (∈, ∈ ∨qk)-fuzzy left ideal ofSif and only iffor anyx,y ∈ Sand α ∈ (0,1], the following conditions hold.
(F1d)
(F2d)
(⟸) Let f be an (∈, ∈ ∨qk-fuzzy left ideal of S and let x, y ∈ S and α∈(0,1] be such that xα ∈ f. Then f(x) ⩾ α. By Proposition 3.2, we have
(⟸) Assume that (F1d) and (F2d) hold. Let x,y ∈ S and set
In the following, we intend to construct an (∈, ∈ ∨qk-fuzzy left ideal from an arbtrary fuzzy subset of a fuzzy ordered semigroup.
LetSbe a fuzzy ordered semigroup andf ∈ F(S). Then
Straightforward by Propositions 2.10 and 3.5.
LetSbe a fuzzy ordered semigroup andf ∈ F(S) . Put
We will complete the proof by three steps.
Step 1: 〈f〉 is an (∈, ∈ ∨qk-fuzzy left ideal of S. For any x,y ∈ S, define two sets
Let
Step 2:
This implies f ⊆ 〈f〉, as required.
Step 3: 〈f〉 ⊆ g for any (∈,∈ ∨ qk-fuzzy left ideal g of S with f ⊆ g. For any
This 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.
Letfbe any fuzzy subset of S. Then
For x,y ∈ S, define
Now, we split the proof into three parts as follows.
(i) i(f) is an (∈, ∈ ∨qk-fuzzy left ideal of S. For any
(ii)
(iii) g ⊆ i(f) for any (∈, ∈ ∨qk-fuzzy left ideal g that is contained (under ⊆ ∨qk) in f. For any fuzzy point xα ∈ g, since g is an (∈, ∈ ∨qk)-fuzzy left ideal, it follows from Proposition 3.6 that
This implies g ⊆ i(f), as required.
The following two propositions are easy to prove.
Let {fi : i ∈ Ibe a family of (∈, ∈ ∨qk)-fuzzy left ideals of a fuzzy ordered semigroup S. Then
Letfandgbe two (∈, ∈ ∨qk)-fuzzy left ideals of afuzzy ordered semigroup S. Then so is
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
Let us recall that a quantale is a triple (Q,*,⩽) such that (Q, ⩽) is a complete lattice, (Q,*) is a semigroup and for any x ∈ Q and {yi}i∈I ⊆ Q,
Combing Proposition 2.10, 3.10 and 3.11, we obtain the following theorem.
LetSbe a fuzzy ordered semigroup. Then (Fidl(S), ⊆, ∘e) is a quantale.
4 (∈, ∈ ∨qk)-fuzzy interior ideals
LetSbe a fuzzy ordered semigroup andf ∈ F(S) . Thenfis called an (∈, ∈ ∨qk-fuzzy interior ideal ofSif it satisfies(Fla)and for anyx,y,z ∈ Sand α, β ∈ (0,1], the following conditions hold.
LetSbe a fuzzy ordered semigroup andf ∈ F(S) . Thenfis an (∈, ∈ ∨qk-fuzzy interior idealofSif and only if
The proof is similar to that of Proposition 3.2. □
Let(S, e, ⋅)be a fuzzy ordered semigroup andf ∈ F(S). Thenf is an (∈, ∈ ∨qk)-fuzzy interior ideal of S if and only if for each
The proof runs parallel to that of Proposition 3.4. □
LetS be a fuzzy ordered semigroup and f ∈ F(S). Then f is an (∈, ∈ ∨qk-fuzzy interior ideal of S if and only if
(F1c)
The proof is similar to that of Proposition 3.5. □
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)
The proof runs parallel to that of Proposition 3.6. □
Let S be a fuzzy ordered semigroup and f ∈ F(S). Put
The proof is similar to that of Theorem 3.8. □
Let
It is easy to verify. □
The following example indicates that
Consider the fuzzy ordered semigroup
Let f and g be two fuzzy subsets of S such that
Then, 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
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
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.
([35]). A non-empty family
(1)
(2)
(3)
As a fuzzy version of above definition, we present the following.
Let
(1)
(2)
(3)
Let
Let
Clearly, by Proposition 4.7, the items (1) and (2) in Definition 4.10 hold in
Now, we prove that Equation (¶) holds. It is clear that
(iii) For any x, y, z ∈ Q, we have
Therefore,
Let
Let L be a complete lattice. An element x ∈ L is said to be compact, if for any subset
([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.
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
Then it is easy to veri that gn is an (∈, ∈ ⋅qk)-fuzzy interior ideal of S for every
Let (S, ⩽, ⋅) be an ordered semigroup. Given any chain of subsets
Let
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, and
Hence, by the hypothesis, fα is an interior ideal of ordered semigroup (S ⩽α #183;) for every
Concerning the relationship between (∈,∈ ∨ qk)-fuzzy ideals and (∈,∈ ∨ qk)-fuzzy interior ideals of a fuzzy ordered semigroup, we have the following result.
Let(S,e,·)be a fuzzy ordered semigroup. Then each (∈,∈ ∨ qk)-fuzzy ideal ofSis an (∈,∈ ∨ qk)-fuzzy interior ideal ofS.
It is straightforward.
The converse of Proposition 4.16 is not true in general as shown in the following example.
Consider the fuzzy ordered semigroupS={a,b,c,d}, where fuzzy ordereand multiplication · aredefined respectively as follows:
Letfbe a fuzzy subsets ofSsuch that
Thenfis an $(\in_{\mathrm{D}}\in\ \vee q_{k})$-fuzzy interior ideal but not an (∈,∈ ∨ qk)-fuzzy ideal ofSfor any $k\in[0,1$), because
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 x ∈ S, there exist a,b,c ∈ S such that x ⩽ axbxc. Using fuzzy partial orders, we can now develop a more generalized type of semisimple ordered semigroups.
A fuzzy ordered semigroup(S,e,·)is said to be generalized semisimple, if for anyx ∈ S,
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.
A fuzzy ordered semigroup(S,e,·)is generalized semisimple if and only iffor any
The proof is straightforward by Definition 5.1.
Next we answer the question that we proposed in the previous section.
LetSbe a generalized semisimple fuzzy ordered semigroup. Then every (∈,∈ ∨ qk)-fuzzy interiorideal is an $(\in,\ \in\vee q_{k})$-fuzzy ideal ofS.
Let f be an (∈,∈ ∨ qk)-fuzzy interior ideal of S and x,y ∈ S. Then it follows from Propositions 3.2 and Definition 5.1 that
Similarly, we can prove that
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.
A fuzzy ordered semigroup(S,e,·)is generalized semisimple if and only if
(⟸) Suppose S is a fuzzy ordered semigroup. Let x be any element of S. Then it is easy to verify that
where the last equality follows from the fact that
Then, from Lemma 2.1 it follows that
Further, by the item (2) of Proposition 2.12, we have
In a similar way, we can prove that
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
implying that
A fuzzy ordered semigroupSis generalized semisimple if and only iffor any two (∈,∈ ∨ qk)-fuzzyinterior idealsfandgofS,
(⟹) 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
This proves that
(⟸) 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
Combining Propositions 5.3 and Theorem 5.5, we obtain the following theorem.
Sbe a fuzzy ordered semigroup. Then the following conditions are equivalent.
(1) Sis generalized semisimple.
(2)
(3)
(4)
(5)
6) The set of all (∈, ∈∨ qk-fuzzy ideals ofSforms a semilattice under the multiplication ∘eand the relation ≡konF(S), that is,
(7) The set of all (∈,∈ ∨ qk)-fuzzy interior ideals ofSforms a semilattice under the multiplication ∘eand therelation ≡konF(S), that is,
LetSbe a fuzzy ordered semigroup. Then the following conditions are equivalent.
(1) Sis generalized semisimple.
(2)
(3)
(4)
(5)
Assume that (1) holds. Let x be an element of S. Then
This implies that
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
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).
References
[1] Rosenfeld A., fuzzy groups, J. Math. Anal. Appl., 1971, 35, 512–517.10.1016/0022-247X(71)90199-5Search in Google Scholar
[2] Pu PM., Liu YM., Fuzzy topology I: Neighourhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl., 1980, 76, 571–59910.1016/0022-247X(80)90048-7Search in Google Scholar
[3] Bhakat S.K., Das P., On the definition of a fuzzy subgroup, Fuzzy Set Syst., 1992, 51, 235–241.10.1016/0165-0114(92)90196-BSearch in Google Scholar
[4] Davvaz B., (∈, ∈ ⋅qk)-fuzzy subnearrings and ideals, Soft Comput., 2006, 10, 206–211.10.1007/s00500-005-0472-1Search in Google Scholar
[5] Jun YB., Song S.Z., Generalized fuzzy interior ideals in semigroups, Inform. Sci. 2006, 176, 3079–3093.10.1016/j.ins.2005.09.002Search in Google Scholar
[6] Kazanci O., Yamak S., Generalized fuzzy bi-ideals of semigroups, Soft computing, 2008, 12, 1119–1124.10.1007/s00500-008-0280-5Search in Google Scholar
[7] Khan A., Shabir M., (α, β)-fuzzy interior ideals in ordered semigroups, Lobachevskii J. Math., 2009, 30, 30–39.10.1134/S1995080209010053Search in Google Scholar
[8] Yin YO., Huang X.K., Xu D.H., Li F., The characterizations of h-semisimple hemirings, Int. J. Fuzzy Syst., 2009, 11, 116–122.Search in Google Scholar
[9] Zhan J.M., Yin YO., A new view of fuzzy k-ideals of hemirings, Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology 2012, 23, 169–176.10.3233/IFS-2012-0506Search in Google Scholar
[10] Davvaz B., Khan A., Characterizations of regular ordered semigroups in terms of (α, β)-fuzzy generalized bi-ideals, Inform. Sci., 2011, 181, 1759–1770.10.1016/j.ins.2011.01.009Search in Google Scholar
[11] Jun YB., Generalization of (∈, ∈ ⋅qk)-fuzzy subalgebras in BCK/BCI-algebras, Comput. Math. Appl., 2009, 58, 1383–1390.10.1016/j.camwa.2009.07.043Search in Google Scholar
[12] Jun YB., Dudek W.A., Shabir M., Kang M.S., General types of (α, β)-fuzzy ideals of hemirings, Honam Math. J., 2010, 32(3), 413–439.10.5831/HMJ.2010.32.3.413Search in Google Scholar
[13] Shabir M., Jun YB., Nawaz Y., Semigroups characterized by (∈, ∈ ⋅qk)-fuzzy ideals, Comput. Math. Appl., 2010, 60, 1473–1493.10.1016/j.camwa.2010.06.030Search in Google Scholar
[14] Jun YB., Meng J., Xin X.L., On ordered filters of implicative semigroups, Semigroup Forum, 1997, 54, 75–82.10.1007/BF02676588Search in Google Scholar
[15] Kehayopulu N., Tsingelis M., Fuzzy sets in ordered groupoids, Semigroup Forum, 2002, 65, 128–132.10.1007/s002330010079Search in Google Scholar
[16] Kehayopulu N., Tsingelis M., Fuzzy bi-ideals in ordered semigroups, Inform. Sci., 2005, 171, 13–28.10.1016/j.ins.2004.03.015Search in Google Scholar
[17] Kehayopulu N., Tsingelis M., Fuzzy interior ideals in ordered semigroups, Lobachevskii J. Math., 2006, 21, 65–71.Search in Google Scholar
[18] Kehayopulu N., Tsingelis M., Regular ordered semigroups in terms of fuzzy subsets, Inform. Sci., 2006, 176, 3675–3693.10.1016/j.ins.2006.02.004Search in Google Scholar
[19] Yin YO., Zhan J.M., Characterization of ordered semigroups in terms of fuzzy soft ideals, Bull. Malays. Math. Sci. Soc., 2012, 4(4): 1–26Search in Google Scholar
[20] Shabir M., Khan A., On fuzzy ordered semigroups, Inform. Sci., 2014, 274, 236–248.10.1016/j.ins.2014.02.107Search in Google Scholar
[21] Xie X.Y, Ideals in lattice-ordered semigroups, Soochow Math. J., 1996, 2275–84.Search in Google Scholar
[22] Xie X.Y, An Introduction to Ordered Semigroup, Publishing Company for Science, Peking, 2001 (in Chinese).Search in Google Scholar
[23] Zadeh L.A., Similarity relations and fuzzy orderings, Inform. Sci., 1971, 3, 177–200.10.1016/S0020-0255(71)80005-1Search in Google Scholar
[24] Zhang Q.Y, Fan L., Continuity in quantitative domains, Fuzzy Set Syst., 2005, 154(1), 118–131.10.1016/j.fss.2005.01.007Search in Google Scholar
[25] Zhang Q.Y, Xie W.X., Fan L., Fuzzy complete lattices, Fuzzy Set Syst., 2009, 160(16), 2275–2291.10.1016/j.fss.2008.12.001Search in Google Scholar
[26] Lai H.L., Zhang D.X., Fuzzy preorder and fuzzy topology Fuzzy Set Syst., 2006, 157(14), 1865–1885.10.1016/j.fss.2006.02.013Search in Google Scholar
[27] Yao W., Shi F.G., A note on specialization L-preorder of L-topological spaces, L-fuzzifying topological spaces, and L-fuzzy topological spaces, Fuzzy Set Syst., 2008, 159, 2586–2595.10.1016/j.fss.2008.03.023Search in Google Scholar
[28] Hao J., Li O.G., The relationship between L-fuzzy rough set and L-topology, Fuzzy Set Syst., 2011, 178, 74–83.10.1016/j.fss.2011.03.009Search in Google Scholar
[29] Yao W., Quantitative domains via fuzzy sets: Part I: Continuity of directed complete posets, Fuzzy Set Syst., 2010, 161, 973–987.10.1016/j.fss.2009.06.018Search in Google Scholar
[30] Yao W., Shi F.G., Quantitative domains via fuzzy sets: Part II: Fuzzy Scott topology on fuzzy directed-complete posets, Fuzzy Set Syst., 2011, 173, 60–80.10.1016/j.fss.2011.02.003Search in Google Scholar
[31] Hao J., The research on pomoind actions and fuzzy rough sets, PhD thesis, Hunan University, 2011 (in Chinese).Search in Google Scholar
[32] Wang K.Y., Some researches on fuzzy domains and fuzzy quantales, PhD thesis, Shaanxi Normal University, 2012 (in Chinese).Search in Google Scholar
[33] Lidl R., Pilz G., Applied abstract algebra, second ed., Springer, New York, 1998.10.1007/978-1-4757-2941-2Search in Google Scholar
[34] Fan L., A new approach to quantitative domain theory, Electron. Notes Theoret. Comput. Sci., 2001, 45, 77–87.10.1016/S1571-0661(04)80956-3Search in Google Scholar
[35] Davey B.A., Priestley H.A., lntroduction to Lattices and Order, Cambridge University Press, Cambridge, 2002.10.1017/CBO9780511809088Search in Google Scholar
© Huang and Li, published by De Gruyter Open
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.