Abstract

In this paper, the notions of fuzzy prime ideals and maximal fuzzy ideals of universal algebras are introduced by applying the general theory of algebraic fuzzy systems.

1. Introduction

The commutator (or the product) of ideals and of a ring , written as , is the ideal of generated by all products and , with and ; i.e., In 1984, H. P. Gumm and A. Ursini [1] have studied the commutator (or the product) of ideals in a more general context. They have defined and characterized the commutator of ideals in universal algebras by the use of commutator terms. Later on, A. Ursini [2] applied this product to study prime ideals of universal algebras. P. Agliano [3] then studied the prime spectrum of universal algebras.

The concept of fuzzy sets was first introduced by Zadeh [4] and this concept was adapted by Rosenfeld [5] to define fuzzy subgroups. Since then, many authors have been studying fuzzy subalgebras of several algebraic structures (see [6–9]). As suggested by Gougen [10], the unit interval is not sufficient to take the truth values of general fuzzy statements. U. M. Swamy and D. V. Raju [11, 12] studied the general theory of algebraic fuzzy systems by introducing the notion of a fuzzy subset of a set corresponding to a given class of subsets of having truth values in a complete lattice satisfying the infinite meet distributive law. Swamy and Swamy [13] defined the commutator (or the product) of fuzzy ideals and of a ring as follows: for all . They have used this commutator to define fuzzy prime ideals of rings.

In [14], we have studied fuzzy ideals in universal algebras having a definable constant denoted by , where is a complete distributive lattice satisfying the infinite meet distributive law. We gave a necessary and sufficient condition for a class of algebras to be ideal-determined. In the present paper, we define the commutator of fuzzy ideals in universal algebras and investigate some of its properties. Moreover, we study fuzzy prime ideals and maximal fuzzy ideals in universal algebras as a generalization of fuzzy prime ideals in those well-known structures: in semigroups [15], in rings [13], in semirings [16], in ternary semirings [17], in -rings [18], in modules [19], in lattices [9], and in other algebraic structures.

2. Preliminaries

This section contains some definitions and results which will be used in this paper. For those elementary concepts on universal algebras we refer to [20, 21]. Throughout this paper , where is a class of algebras of a fixed type , and we assume that there is an equationally definable constant in all algebras of denoted by . For a positive integer , we write to denote the tuple .

Definition 1 ([1]). A term is said to be an ideal term in if and only if .

Definition 2 ([1]). A nonempty subset of is called an ideal of if and only if for all and any ideal term in .

We denote the class of all ideals of , by .

Definition 3 ([1]). A class of algebras is called ideal-determined if every ideal is the zero congruence class of a unique congruence relation denoted by . In this case the map defines an isomorphism between the lattice of ideals and congruences on .

Definition 4 ([1, 2]). A term is said to be a commutator term in if and only if it is an ideal term in and an ideal term in .

Definition 5 ([1]). In an ideal-determined variety, the commutator of ideals and is the zero congruence class of the commutator congruence .

It is characterized in [1] as follows.

Theorem 6 ([1, 2]). In an ideal-determined variety,

For subsets of , denotes the product . In particular, for , is denoted by .

Definition 7 ([2]). A proper ideal of is called prime if and only if for all

Theorem 8 ([2]). A proper ideal of is prime if and only if for all .

Throughout this paper is a complete Brouwerian lattice; i.e., is a complete lattice satisfying the infinite meet distributive law. By an fuzzy subset of , we mean a mapping . For each , the level set of denoted by is a subset of given by the following. For fuzzy subsets and of , we write to mean in the ordering of .

Definition 9 ([22]). For each and in , the fuzzy subset of given by is called the fuzzy point of . In this case is called the support of and is its value.

For an fuzzy subset of and an fuzzy point of , we write whenever .

Definition 10 ([14]). An fuzzy subset of is said to be an fuzzy ideal of if and only if the following conditions are satisfied: (1), and(2)If is an ideal term in and ; then

Note that an fuzzy subset of satisfying the two conditions in the above definition can be regarded as a normal fuzzy ideal in the sense of Jun et al. [23].

Theorem 11 ([14]). A class of algebras is ideal-determined if and only if every fuzzy ideal is the zero fuzzy congruence class of a unique fuzzy congruence relation denoted by .

3. The Commutator of Fuzzy Ideals

In this section, we define the commutator (or the product) of fuzzy ideals in universal algebras. It is observed in [14] that an fuzzy subset of is an fuzzy ideal of if and only if every level set of is an ideal of . Here we define the commutator of fuzzy ideals using their level ideals.

Definition 12. The commutator of fuzzy ideals and of denoted by is an fuzzy subset of defined by for all .

For each , and in with , the following can be verified. So the commutator of fuzzy ideals can be equivalently redefined as follows. The following lemmas can be verified easily.

Lemma 13. For any fuzzy ideals and of , is an fuzzy ideal of such that

Lemma 14. For any ideals and of

In the following theorem, we give an algebraic characterization for the commutator of fuzzy ideals.

Theorem 15. For each and fuzzy ideals and of

Proof. For each , let us define two sets and as follows. Clearly both and are subsets of . Our claim is to see the following. One way of proof is to show that . implies that , where for some , and some commutator term in . That is, and so that . Then and hence . Another way is to prove that, for each , there exists such that . If , then so that for some , and , where is a commutator term in . That is, and . If we put , then and . This completes the proof.

Notation 16. We write , to say that is a finite subset of .

Theorem 17. For each and fuzzy ideals and of

Proof. For each , let us take the set as in Theorem 15 and define a set as follows. Our claim is to show the following. One way of proof is to show that . If , then where and are finite subsets of such that . That is, for all and all . Then and so that . Thus and hence . Therefore . Another way is to prove that, for each , there exists such that . If , then . Therefore, for some , and , where is a commutator term in . That is, and If we put and , then and are both finite subsets of such that . Moreover, if we take then such that . This completes the proof.

Definition 18. For each , we define by induction An fuzzy ideal of will be called fuzzy nilpotent (resp., fuzzy solvable) if (resp., ) for some .

Lemma 19. An fuzzy subset of is fuzzy nilpotent (resp., fuzzy solvable) if and only if is nilpotent (resp., solvable) for all .

4. Fuzzy Prime Ideals

In this section we define fuzzy prime ideals and investigate some of their properties.

Definition 20. A nonconstant fuzzy ideal of is called an fuzzy prime ideal if and only if for all .

Notation 21. For fuzzy points and of , we denote by .

In the following theorem we characterize fuzzy prime ideals using fuzzy points.

Theorem 22. A nonconstant fuzzy ideal of is fuzzy prime if and only if for any fuzzy points and of

Proof. Suppose that satisfies the condition: for all fuzzy points and of . Let and be fuzzy ideals of such that . Suppose if possible that and . Then there exist such that and . If we put and , then and are fuzzy points of such that , but , and , but , so that , but and . This contradicts our hypothesis. Thus either or . Therefore is prime. The other way is clear.

Theorem 23. A nonconstant fuzzy ideal is an fuzzy prime ideal if and only if , where is a prime element in and the set is a prime ideal of .

Proof. Suppose that is a prime fuzzy ideal. Clearly , and since is nonconstant, there is some such that . We show that for all . Let such that and . Let us define fuzzy subsets and of as follows: and for all . Then it can be verified that both and are fuzzy ideals of . Moreover, for each we have so that , but , so . Since is fuzzy prime, we get , which gives ; that is, . Similarly it can be verified that so that for all . Thus for some in . It remains to show that is a prime element in . Let such that . Consider fuzzy subsets and of defined by and for all . Then and are both fuzzy ideals of such that . Since is fuzzy prime, either or so that either or . Hence is prime in . Next we show that the level ideal is prime. Put and let and be ideas of such that . Then . That is, . Since is fuzzy prime, either or , implying that either or . Therefore is prime. Conversely suppose that , where is a prime element in and is a prime ideal of . Let and be fuzzy ideals of such that . Suppose if possible that there exist such that and . Since is valued, so that both and do not belong to . Since is prime, there exists such that ; that is, . Otherwise, if , then either or . As , for some , , where is a commutator term in . Now consider the following. That is, . Since is a prime element in , it follows that either or , which is a contradiction. Therefore is fuzzy prime.

Let be a prime ideal of and be a prime element in . Consider an fuzzy subset of defined by for all . The above theorem confirms that fuzzy prime ideals of are only of the form . This establishes a one-to-one correspondence between the class of all fuzzy prime ideals of and the collection of all pairs where is a prime ideal in and is a prime element in .

Corollary 24. Let be an ideal of and a prime element in . Then is a prime ideal if and only if is an fuzzy prime ideal.

Theorem 25. If is an fuzzy prime ideal of , then for all .

Proof. We use proof by contradiction. Suppose if possible the following. Then for all . Since is prime, by Theorem 23 there exists a prime element in such that and for all , so . Since is a prime ideal of (see Theorem 23), we get that either or . This is a contradiction. Thus the result holds.

Theorem 26. If , where is a prime element in and satisfies the condition: for all a, then is fuzzy prime.

It is natural to ask ourselves, does every algebra in have fuzzy prime ideals? Of course, probably no. In the following theorem we give a sufficient condition for an algebra to have fuzzy prime ideals.

Theorem 27. Let be an algebra satisfying the following. If and is an fuzzy ideal of such that where is an irreducible element in , then there exists an fuzzy prime ideal of such that

Proof. Put . Clearly so that is nonempty and hence it forms a poset under the inclusion ordering of fuzzy sets. By applying Zorn’s lemma we can choose a maximal element, say , in . Now it is enough to show that is prime. Suppose not. Then there exist fuzzy ideals and of such that but and . Put and . Then and are fuzzy ideals of such that and . By the maximality of in both and do not belong to . Thus Since and is irreducible element in , we get , so , which is a contradiction. Therefore is prime.

If is a nontrivial algebra such that for all , then it can be deduced from the above theorem that fuzzy prime ideals exist in .

Definition 28. An fuzzy subset of is called an fuzzy system (resp., an fuzzy system) if for all , there exists (resp., there exists ) such that

Lemma 29. An fuzzy subset of is an fuzzy system (resp., a fuzzy system) if and only if the level set is an system (resp., an system) for all .

Theorem 30. Let be an fuzzy ideal of and an fuzzy system such that , where is an irreducible element in . Then there exists fuzzy prime ideal of such that

Proof. Put . Clearly so that is nonempty, and hence it forms a poset under the inclusion ordering of fuzzy sets. By applying Zorn’s lemma we can choose a maximal element, say , in . Now it is enough to show that is prime. Suppose not. Then there exist fuzzy ideals and of such that , but and . Put and . Then and are fuzzy ideals of such that and . By the maximality of in both and do not belong to so there exist such that Since , we have . If , then for some , , , and some commutator term in . Then for each the following holds. Also we have , which gives that . However, since and is an irreducible element in , we get that for all . This contradicts that is a fuzzy system. Therefore is prime.

For a nontrivial algebra , to have an fuzzy system is a sufficient condition for to possess fuzzy prime ideals.

5. Maximal Fuzzy Ideals

A maximal fuzzy ideal of is a maximal element in the collection of all nonconstant fuzzy ideals of under the pointwise partial ordering of fuzzy sets.

An element in is called a dual atom if there is no in such that . In other words is maximal in . In the following theorem we give an internal characterization of fuzzy maximal ideals in .

Theorem 31. An fuzzy ideal of is maximal if and only if , where is a dual atom in and the set is a maximal ideal of .

Proof. Suppose that is maximal. Clearly , and since is nonconstant, there is some such that . We first show that assumes exactly one value other than . Let such that and . Put and . Define fuzzy subsets and of as follows: for all . Then it can be verified that both and are fuzzy ideals of such that and . By the maximality of we get that and . Thus . Therefore for some . Next we prove that this is a dual atom. Let such that . Define an fuzzy subset of by for all . Then is an fuzzy ideal of such that . By the maximality of it yields that ; i.e., for all , so . Therefore is a dual atom. It remains to show that is a maximal ideal of . Clearly it is a proper ideal. Let be a proper ideal of such that . Define an fuzzy subset of by for all . Then is a nonconstant fuzzy ideal of such that . Since is maximal, we get , so . Therefore is maximal among all proper ideals of . Conversely suppose that , where is a dual atom in and the set is a maximal ideal of . Let be a nonconstant fuzzy ideal of such that . Then for all and for all . We show that . Suppose not. Then there exists such that , so . If , then and . This contradicts the maximality of . Also if , then . Again this contradicts the hypothesis that is a dual atom. Therefore . Hence is maximal.

The above theorem confirms that there is a one-to-one correspondence between the class of all maximal fuzzy ideals and the set of all pairs where is a maximal ideal in and is a dual atom in .

Theorem 32. If is an algebra in which every maximal ideal is a prime ideal, then every maximal fuzzy ideal is an fuzzy prime ideal.

For instance, if , then every maximal fuzzy ideal is an fuzzy prime ideal.

6. Conclusion

In this paper, the commutator (or the product) of fuzzy ideals is defined and investigated in a more general context, in universal algebras. By the use of this commutator, fuzzy prime ideals in universal algebras are defined and fully characterized. Furthermore, maximal fuzzy ideals of universal algebra are studied.

The study of fuzzy semiprime ideals in universal algebras is under investigation by the authors using the commutator of fuzzy ideals. Moreover, the radical of fuzzy ideals in universal algebras will be studied and would be applied to characterize the central properties of fuzzy semiprime ideals.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research work was partially supported by a research grant from the College of Science, Bahir Dar University.