Research Article | Open Access

# On Fuzzy Congruences and Fuzzy Strong -Ideals of Hemirings

**Academic Editor:**Anca Croitoru

#### Abstract

The aim of this paper is to introduce the concepts of fuzzy strong *h*-ideals and fuzzy congruences of hemirings. The quotient hemirings via fuzzy strong *h*-ideals are investigated. The relationships between fuzzy congruences and fuzzy strong *h*-ideals of hemirings are discussed. Pay attention to an open question on fuzzy congruences. Finally, the normal fuzzy strong *h*-ideals of hemirings are explored.

#### 1. Introduction

The concept of fuzzy sets was formulated by Zadeh [1], and since then there has been a remarkable growth of fuzzy set theory. In 1989, Filep and Maurer [2] introduced the concepts of fuzzy congruences and compatible partitions. Further, Zadeh [3], Murali [4], and Kuroki [5] discussed the properties of fuzzy congruences. Moreover, Dutta and Biswas [6] redefined fuzzy equivalent relations and fuzzy congruences of semirings.

Semirings, regarded as a generalization of rings, have been recently found particularly useful in solving problems in different disciplines of applied mathematics and information sciences because semirings provide an algebraic framework for modelling. A special semiring with a zero and endowed with the commutative addition is said to be a hemiring. Nowadays, semirings (hemirings) are useful in optimization theory, graph theory of discrete event dynamical systems, matrices, determinants, generalized fuzzy computation, automata, theory, formal language theory, coding theory, and analysis of computer programs.

We know that ideal theory of semirings plays a central role in the structure theory and is useful for many purposes. The properties of -ideals and -ideals of hemirings were thoroughly investigated by la Torre [7]. Further, fuzzy -ideals of semirings were investigated by [8–12]. In 2004, Jun et al. [13] introduced the concept of fuzzy -ideals of hemirings and investigated some related properties of hemirings. In particular, the -hemiregular hemirings were described by Zhan and Dudek [14]. Now, many researchers investigated hemirings, such as Ma et al. [15–20].

In this paper, we study fuzzy congruences and fuzzy strong -ideals of hemirings. The remaining part of this paper is organized as follows. In Section 2, we first recall some basic definitions of hemrings and give the concepts of strong -ideals and fuzzy strong -ideals of hemirings. In Section 3, we consider fuzzy homomorphisms of hemirings and the quotient hemirings via fuzzy strong -ideals. In Section 4, we investigate the relationships between fuzzy congruences and fuzzy strong -ideals. In Section 5, we introduce normal fuzzy strong -ideals of hemirings.

#### 2. Preliminaries

Recall that a semiring is an algebraic system consisting a nonempty set of together with two binary operations on called an addition and a multiplication (denoted in the usual manner) such that and are semigroups and the following distributive laws: and are satisfied for all .

By zero of a semiring we mean an element such that and for all . By an identity of a semiring we mean an element such that for all . A semiring with a zero and a commutative semiring is called a hemiring. Throughout this paper, is always a hemiring.

A subset of is called a left(right) ideal of if is closed under addition and . A left ideal of is called a left -ideal if for any , and and , it follows . A right -ideal is defined analogously. A left ideal of is called a strong left -ideal if for any and from , it implies . A strong right -ideal is defined analogously. Every strong -ideal is an -ideal; the converse is not true [20].

A fuzzy set of is defined as a mapping from to the real interval [0, 1]. Let denote the set of all fuzzy sets of . A fuzzy set in of the form is called a fuzzy point with support and value and is denoted by . In particular, if , denotes the fuzzy point with support and value .

*Definition 1 (see [13]). *A fuzzy set of is called a fuzzy left ideal if for all , we have,.A fuzzy left ideal of is called a fuzzy left -ideal if for all . A fuzzy right -ideal is defined similarly.

*Definition 2 (see [20]). *Let and be fuzzy sets of . The sum of and is defined by

In particular,

Note that if and , then .

*Definition 3. *A fuzzy set of is called a fuzzy strong left(right) -ideal of if for all ,(1),
(2),
(3). Note that if is a fuzzy strong -ideal of , then .

*Example 4. *Let be a set with an addition operation and a multiplication operation as follows:

Define a fuzzy set in by . Then is a fuzzy strong -ideal of .

*Definition 5 (see [18]). *Let and be fuzzy sets of . Then -sum of and is defined by
if there exist such that ; otherwise, .

Let be a fuzzy set of and . Then the sets and are called an -level subset and an -strong level of , respectively. We now characterize the fuzzy strong -ideal of by using their (strong) level subsets.

Theorem 6. *A fuzzy set of is a fuzzy strong -ideal of if and only if nonempty is a strong -ideal of for all .*

*Proof. *Let be a fuzzy strong -ideal of , , and . Then , and so . Similarly, we get , hence is an ideal of .

Now, let and be such that . Hence, . Thus we have

This implies that there exists such that and ; that is, , and so . Therefore, is a strong -ideal of .

Conversely, assume that the given conditions hold. Let . If possible, let . Choose such that . Then , but , a contradiction. Hence, for all . Similarly, we have for all .

Now assume there exist such that and ; choose such that . Then , but . That is, , a contradiction. Hence, for all with . This implies that is a fuzzy strong -ideal of .

#### 3. Quotient Hemirings and Their Isomorphisms

In this section, the quotient hemirings via fuzzy strong -ideals are investigated. Finally, we give an isomorphism theorem of hemirings.

Let be an equivalence relation on . Recall that is called a congruence relation on if and imply and .

Let be a strong -ideal of , . We call congruent to mod , if and only if there exist and be such that [20]. It is checked that the relation is a congruence relation on .

Lemma 7 (see [20]). *Let be a strong -ideal of . If , then*(1)* if and only if ,*(2)*,
*(3)*. **Let be a fuzzy strong -ideal of and an -level subset of . We denote by the equivalence class containing and by the set of all equivalence classes of .*

Theorem 8. *If is a fuzzy strong -ideal of and , then is a hemiring under the binary operations:
**
for any .*

*Proof. *Firstly, we show that the above binary operations are well defined. In fact, if and . Since , we have , and so . Then . By Lemma 7, we have , and so ; that is, . Hence the addition is well defined.

By Lemma 7, we also know . Next we show . Let ; since , then . This implies that , and so ; that is, . This means that . Hence the multiplication is well defined. Now it is easy to verify that is a hemiring.

*Definition 9 (see [21]). *Let be a homomorphism from to , a fuzzy subset of , and a fuzzy subset of . Then the image of and the preimage of are both fuzzy sets defined, respectively, as follows:
for all .

*Definition 10. *Let be a homomorphism of hemirings. A strong -ideal of is called -compatible, if for all , implies . A fuzzy strong -ideal of is called -compatible, if for all , implies .

*Remark 11. *If the above is a monomorphism, then every fuzzy strong -ideal is -compatible.

Theorem 12. *Let be an epimorphism of hemirings. If is a -compatible fuzzy strong -ideal of , then is a fuzzy strong -ideal of .*

* Proof. *(1) Let ; then

(2) Let ; then

(3) Let and be such that ; then we have . Since is -compatible, we have ; thus
and so

Therefore, is a fuzzy strong -ideal of .

Similarly, we can obtain the following result.

Theorem 13. *Let : be a monomorphism of hemirings. If is a fuzzy strong -ideal of , then is an -compatible fuzzy strong -ideal of .*

*Proof. *The proof is similar to Theorem 12.

Theorem 14. *Let and be two fuzzy sets of . If they are fuzzy strong left (resp., right) -ideals of , then so are and .*

*Proof. *We first show that is a fuzzy strong left -ideal of . For any ,

Since and , it follows that

Therefore, is a fuzzy left ideal of .

Let be such that . Then

Hence is a fuzzy strong left -ideal of .

Now we show that is a fuzzy strong left -ideal. In fact,

(1) for any , we have

(2) For any , we have

(3) Let be any elements of such that . If there exist , such that
then we have
where , and so

This gives
Otherwise, we have or , and so . Summing up the above statements, is a fuzzy strong left -ideal of . The case for fuzzy strong right -ideals can be similarly proved.

Now denote by the set of all fuzzy strong -ideals of with the same tip ; that is, for all . Then we have the following result.

Theorem 15. *The is a bounded complete lattice under the relation “”.*

* Proof. *Let . It follows from Theorem 14 that and . It is clear that is the greatest lower bound of and . We now show that is the least upper bound of and , since ; for any , we have

Hence, . Similarly, we have . Now, let be such that ; then we have . Hence, . It is clear that replace the with arbitrary family of and so is a complete lattice under the relation “”. and are the minimal and the maximal elements in , respectively. Therefore, is a bounded complete lattice.

Finally, we give an isomorphism theorem of hemirings.

Theorem 16. *Let : be an isomorphism of hemirings and a fuzzy strong -ideal of . If and , then
*

* Proof. *It follows from Theorems 8 and 13, and are both hemirings. Define by

(1) is well defined as follows: . Since is a homomorphism, then .

(2) is a homomorphism:

(3) is an epimorphism; for any , since is epimorphism, then there exists , such that , so .

(4) is monomorphism: ; since is monomorphism, so ; thus , so we have ; hence , and the proof is completed.

#### 4. The Relationships between Fuzzy Congruences and Fuzzy Strong -Ideals

In this section, we investigate the relationships between fuzzy congruences and fuzzy strong -ideals of hemirings. The following concepts can be seen in [6].

*Definition 17. *A nonempty fuzzy relation on is called a fuzzy equivalence relation if(1) (fuzzy reflexive),(2) (fuzzy symmetric),(3) for all (fuzzy transitive).

*Definition 18. *A fuzzy equivalence relation on is called a fuzzy congruence if

*Example 19. *Consider the set of all nonnegative integers is a hemiring with respect to the usual addition and multiplication; we define a fuzzy relation on as follows.

For all in ,

Theorem 20. *Let be a fuzzy congruence on and let be the fuzzy subset of defined by
**If there exist be such that ; otherwise, . Then is a fuzzy strong -ideal of .*

*Proof. *(1) For any , we have

(2) For any , we have

Similarly, .

(3) Let be any elements of such that ; if there exist be such that
then we have , where , and so

This gives

Otherwise, we have or , and so . Summing up the above statements, is a fuzzy strong -ideal of .

*Remark 21. * is called the fuzzy strong -ideal induced by .

Theorem 22. *Let be a fuzzy strong -ideal of . Let be the fuzzy relation on defined by
**. Then is a fuzzy equivalence relation on .*

*Proof. *Since is nonempty, it follows that is also nonempty. Now
for any in . Again for in . So for any . Hence ; that is . Therefore, is fuzzy reflexive. Obviously is fuzzy symmetric. Now

Hence . Thus, is a fuzzy equivalence relation on .

*Open Question*. Theorem 22 indicates that a fuzzy strong -ideal of can induce a fuzzy equivalence relation on . However, the question whether a fuzzy strong -ideal of can induce a fuzzy congruence on is still open.

#### 5. Normal Fuzzy Strong Left -Ideals

In this section, we introduce the normal fuzzy strong left -ideals of hemirings.

*Definition 23. *A fuzzy strong left -ideal of is said to be normal if there exists such that .

It is obvious to verify that is normal if and only if . We also note that any fuzzy strong -ideal containing some normal strong left -ideals is normal. Since if and is normal, then , so .

Proposition 24. *Given a fuzzy strong left -ideal of , let be a fuzzy set in defined by for all ; then is a normal fuzzy strong left -ideal of which contain .*

*Proof. *Firstly, we prove for all ; we have and

which proves . Then we prove . Similarly,

This proves that holds. Hence is a fuzzy left ideal of . Now, let be such that . Then

Since is a fuzzy strong left -ideal, we have , so

Therefore, is a normal fuzzy strong left -ideal of , and obviously .

Corollary 25. *Let and be as in Proposition 24. If there exists such that , then .*

For any strong left -ideal of , the characteristic is a normal fuzzy strong left -ideal of . It is clear that is a normal fuzzy strong left -ideal of if and only if .

Proposition 26. *If is a fuzzy strong left -ideal of . Then . Moreover, if is normal, then .*

*Proof. *It is straightforward.

Theorem 27. *Let be a fuzzy strong left -ideal of and let be an increasing function. Then a fuzzy set defined by is a fuzzy strong left ideal of . In particular, if , then is normal; if for all , .*

*Proof. *For all , then we have
which proves . Similarly, , which proves holds. Hence is a fuzzy left ideal of .

Now, let be such that . Then

Since is a fuzzy strong left -ideal, we have

so

Therefore, is a fuzzy strong left -ideal of ; if , then is normal; suppose that for all , which proves .

Let denote the set of all normal fuzzy strong left -ideal of . Note that is a poset under the set inclusion.

Theorem 28. *Let be nonconstant such that it is a maximal element of . Then takes only two values 0 and 1.*

*Proof. *Since is normal, we have . For some , let ; we know that ; otherwise, there exists , such that . Now we define on a fuzzy set by putting for some . Then is well defined.

For all , we have
which proves holds.

Similarly, we can obtain
which proves holds. Hence is a fuzzy left ideal of . Now, let be such that ; since is a normal fuzzy strong left -ideal, we have , which implies
Therefore, is a fuzzy strong left -ideal of . By Proposition 24, is a normal fuzzy strong left -ideal of . Note that
and ; this means that is nonconstant and is not a maximal of . This is a contradiction.

*Definition 29. *A nonconstant fuzzy strong left -ideal of is called maximal if is a maximal element of .

Theorem 30. *If a fuzzy strong left -ideal of is maximal, then*(1)* is normal,*(2)* takes only the values 0 and 1,*(3)*,*(4)* is a maximal strong left -ideal of .*

*Proof. *Let be a maximal fuzzy strong left -ideal of . Then is a nonconstant maximal element of the poset . It follows from Theorem 28 that takes only the values 0 and 1. Note that if and only if , and if and only if . By Corollary 25, we have , and so . Hence is normal and . This proves that (1) and (2) hold.

(3) Obviously.

(4) It is clear that