Abstract

The aim of this paper is to investigate the lattices of group fuzzy congruences and normal fuzzy subsemigroups on -inversive semigroups. We prove that group fuzzy congruences and normal fuzzy subsemigroups determined each other in -inversive semigroups. Moreover, we show that the set of group -fuzzy congruences and the set of normal subsemigroups with tip in a given -inversive semigroup form two mutually isomorphic modular lattices for every .

1. Introduction

The investigation of fuzzy sets is initiated by Zadeh in [1]. As special fuzzy sets, fuzzy congruences on groups and semigroups have been extensively studied by many authors. In 1992, Kuroki [2] introduced fuzzy congruences on a group and characterized fuzzy congruences on a group in terms of fuzzy normal subgroups. In 1993, Samhan [3] studied the modularity condition in the fuzzy congruence lattice of a semigroup and derived that the fuzzy congruence lattice of a group is modular. In the same year, Al-Thukair [4] described the fuzzy congruences of an inverse semigroup and obtained a one-one correspondence between fuzzy congruence pairs and fuzzy congruences on an inverse semigroup. Moreover, Kuroki also studied the fuzzy congruences on inverse semigroups in [5] in which the notion of group congruences of a semigroup is provided. Das [6] considered the lattice of fuzzy congruences in an inverse semigroup by kernel-trace approaches. In 1995, Ajmal and Thomas considered the lattice structures of fuzzy congruences on a group and the lattice structures of fuzzy subgroups and fuzzy normal subgroups in a group in [7] and proved that the lattice of fuzzy normal subgroups of a group is modular in [8]. In 1997, Kim and Bae [9] studied the fuzzy congruences of groups and obtained several results which are analogs of some basic theorems of group theory. Also, Xie [10] studied the so-called fuzzy Rees congruences on semigroups in 1999.

Several authors investigated fuzzy congruences for some special classes of semigroups. In 2000, Zhang [11] characterized the group fuzzy congruences on a regular semigroup by some fuzzy subsemigroups. In 2001, Tan [12] investigated fuzzy congruences of regular semigroups and proved that the lattice of fuzzy congruences on a regular semigroup is a disjoint union of some modular sublattices of the lattice. Recently, Li and Liu [13] characterized fuzzy good congruences of left semiperfect abundant semigroups and obtained sufficient and necessary conditions for an abundant semigroup to be left semiperfect.

The class of -inversive semigroups is a very wide class of semigroups which contains groups, inverse semigroups, and regular semigroups as proper subclasses and some kinds of crisp congruences on this class of semigroups have been investigated extensively; see [14, 15] for example. In particular, Gigoń [14] considered the lattice of group crisp congruences on an -inversive semigroup and proved that this lattice is modular. Inspired by the above facts, it is natural to study the fuzzy congruences on -inversive semigroups. In fact, [16] has done some works in this aspect.

In this paper, we shall investigate the lattices of group fuzzy congruences and normal fuzzy subsemigroups on an -inversive semigroup. The notions of group -fuzzy congruences and normal fuzzy subsemigroups with tip on -inversive semigroups are proposed and some properties of them are given. In particular, for a given -inversive semigroup , we prove that for any the set of group -fuzzy congruences and the set of normal fuzzy subsemigroups with tip on form two mutually isomorphic modular lattices. Our results generalize and enrich several results obtained in [2, 3, 8, 9, 11, 14]. Notations and terminologies not given in this paper can be found in [17–19].

2. Preliminaries

A binary relation “” defined on a set is a partial order on the set if the following conditions hold identically in :

  ;    and imply ;    and imply .

Let be a subset of a poset . An element in is an upper bound for if for every in . An element in is the supremum of if is an upper bound of and is the smallest among the upper bounds of . Dually, we can define the infimum of . We denote the supremum and the infimum of by and , respectively.

A poset is called a lattice if for every in both the supremum and the infimum of exist in . A modular lattice is any lattice which satisfies the modular law: implies that for all in .

Two lattices and are isomorphic if there is a bijection from to such that for every in the following two equations hold: and . If and are two posets and is a map from to , then we say is order-preserving if holds in whenever holds in .

On the theory of lattices, we need the following results.

Lemma 1 (Theorem 2.3 of Chapter 1 in [17]). Two lattices and are isomorphic if and only if there is a bijection from to such that both and are order-preserving.

Lemma 2 (Theorem 4.2 of Chapter 1 in [17]). Let be a poset such that it has the largest element and the infimum of every nonempty subset exists. Then is a lattice.

Zadeh [1] defined a fuzzy subset in a set as a mapping from to the closed unit interval . A fuzzy set in a set is said to be contained in a fuzzy set if for all in and this is denoted by . The union and the intersection of two fuzzy sets and in a set are defined by for all in . Further, if is a fuzzy subset in for where is an index set, then is defined by for all .

A semigroup is a nonempty set with an associative binary operation. A semigroup is called E-inversive if for all there exists such that . In this case, we denote and call the elements in the weak inverses of for any . It is easy to see that groups, regular semigroups, and semigroups with zeros are all -inversive semigroups. For more details on -inversive semigroups, see [14, 15] and their references. Throughout this paper, we always assume that is an -inversive semigroup and let

Now, we give the concept of -fuzzy congruences.

Definition 3 (see [19]). Let . A -fuzzy equivalence on is a fuzzy subset in which satisfies the following conditions: (1),(2),(3).A -fuzzy equivalence on is called a - if (4),or, equivalently, (4′).

Similar to the proofs of Lemmas 2.2 and 2.3 in Kuroki [5], we have the following results.

Lemma 4. Let be a -fuzzy congruence on . For any , define a fuzzy subset in as follows: for all .(1) if and only if for all .(2) is a semigroup with the multiplication for any .

3. Group Fuzzy Congruences

In this section, we consider some basic properties of group fuzzy congruences on . In particular, we show that the set of group -fuzzy congruences on forms a modular lattice. Firstly, we give the concept of group -fuzzy congruences which is parallel to that of usual group fuzzy congruences defined in Kuroki [5].

Definition 5. A -fuzzy congruence on is called a group -fuzzy  congruence if the semigroup is a group. One denotes the set of group -fuzzy congruences on by and denotes .

The following result provides a characterization of group -fuzzy congruences on .

Proposition 6. A -fuzzy congruence on is a group -fuzzy congruence if and only if (1);(2).

Proof. If is a group -fuzzy congruence, then is a group and so is the identity of for every . This implies that for all by Lemma 4. Furthermore, since for any and , it follows that is the identity of and so . This yields that by Lemma 4 again.
Conversely, let , , and . Then by condition and Lemma 4, . By condition and Lemma 4, we have which implies that is a group.

Proposition 7. Let , , and . Then

Proof. Since , by Proposition 6, we have for all and . This implies that On the other hand, Therefore, . By similar arguments, we can show for all and .

As usual, for , we define as follows: for all . Then we have the following.

Proposition 8. Let . (1).(2) is the least group -fuzzy congruence of   containing and . (3) is the greatest group -fuzzy congruence of contained in and .

Proof. For all and , by Proposition 7, we have By symmetry, we have for all .
Let . Since we have . Similarly, we have In view of the proofs of Propositions 1.8 and in Kim and Bae [9], is the least -fuzzy congruence on containing and . Finally, we can easily show that by Proposition 6. This implies that is the least group -fuzzy congruence of containing and .
This is clear.

By Proposition 8 and the proof of Theorem 1.12 in Kim and Bae [9], we have the following result.

Theorem 9. forms a modular lattice for any in .

In the end of this section, we give some properties of group -fuzzy congruences on which will be used in the final section.

Proposition 10. Let and . (1) for all . (2) for all and .

Proof. Since , by Proposition 6, we have This yields that whence . Thus, By dual arguments, we can obtain that .
(2) In view of the fact that , by Proposition 7, we have Thus, .

4. Normal Fuzzy Subsemigroups

In this section, we consider some basic properties of normal fuzzy subsemigroups of -inverse semigroups.

Definition 11. Let . A fuzzy subset in is called a normal  fuzzy  subsemigroup  with  tip   in if (1),(2),(3).

We denote the set of normal fuzzy subsemigroups with tip in by and let .

Remark 12. In fact, normal fuzzy subsemigroups with tip are introduced in Zhang [11] where this class of fuzzy subsemigroups is called complete inner-unitary subsemigroups.

Let be a group with identity , and let be a fuzzy set in . From Ajmal and Thomas [8], is called a normal fuzzy subgroup of with tip if for all the following conditions hold: We assert that normal fuzzy subsemigroups with tip are generalizations of normal fuzzy subgroups with tip in the range of -inversive semigroups. To see this, we need the following result.

Proposition 13. Let , , and . (1).(2).(3).

Proof. On the one hand, we have and On the other hand, since , we have . This implies that Therefore, .
This follows from item and the fact that .
The result follows from Proposition 2.6 in Zhang [11].

The following result justifies the name of normal fuzzy subsemigroups.

Theorem 14. Let be a group with identity , and let be a fuzzy subset in . Then is a normal fuzzy subsemigroup with tip of if and only if is a normal fuzzy subgroup of with tip .

Proof. Observe that is the unique idempotent in and the inverse of is certainly the unique weak inverse of for all . If is a normal fuzzy subsemigroup with tip of , then by Proposition 13, is a normal fuzzy subgroup of with tip . Conversely, let be a normal fuzzy subgroup of with tip and . Then by condition (17). This implies that Moreover, we have Thus, is a normal fuzzy subsemigroup of with tip .

Since for , , and the element is the greatest one in , we have the following theorem by Lemma 2.

Theorem 15. forms a lattice.

On normal fuzzy subsemigroups with tip of -inverse semigroups, we also have the following basic properties which will be used in the final section.

Proposition 16. Let , , , and .(1). (2). (3).

Proof. Since , we have . This implies that whence .(2)The result follows from the facts that (3)This follows from the proof of Theorem 2.12 in Zhang [11].

5. The Relationship of and

In this section, we show that is isomorphic to as lattices whence is modular for all in . We first give some useful propositions.

Proposition 17. Let and Then , where .

Proof. In view of Proposition 16, the above is well defined. Now, let and , , . Then we have the following facts:(1)Since , .(2)By Proposition 16, we have
Since , it follows that and by Proposition 13 and Proposition 16. This implies that
For any , we have and by Proposition 13 and Proposition 16, . This implies that . Dually, we have . This implies that Thus, .(5)By Proposition 16 and the fact that , we have .(6).From the above six items, we can see that by Proposition 6.