L-Fuzzy Prime Ideals in Universal Algebras

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 byGougen [10], the unit interval [0, 1] 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 L− subset of a set X corresponding to a given class L of subsets of X having truth values in a complete lattice satisfying the infinite meet distributive law. Swamy and Swamy [13] defined the commutator (or the product) of L−fuzzy ideals μ and σ of a ring R as follows: [μ, σ] (x) = ⋁{ n ⋀


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.,  = { ∈  :  = Σ  =1     ,   ∈ ,   ∈ } .
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][7][8][9]).As suggested by Gougen [10], the unit interval [0, 1] 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 L− subset of a set  corresponding to a given class L 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 0, 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 Γ-rings [18], in modules [19], in lattices [9], and in other algebraic structures.

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  ∈ K, where K is a class of algebras of a fixed type Ω, and we assume that there is an equationally definable constant in all algebras of K denoted by 0. For a positive integer , we write  →  to denote the −tuple ⟨ for all ,  ∈ .
Throughout this paper  = (, ∧, ∨, 0, 1) 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 ([22]).For each  ∈  and 0 ̸ =  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 ([14]).An −fuzzy subset  of  is said to be an −fuzzy ideal of  if and only if the following conditions are satisfied: (1) (0) = 1, and 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].

The Commutator of 𝐿−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 .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.

𝑥 ∈ [𝜇
So the commutator of −fuzzy ideals can be equivalently redefined as follows. [ The following lemmas can be verified easily.
In the following theorem, we give an algebraic characterization for the commutator of −fuzzy ideals.
Our claim is to show the following.
One way of proof is to show that   ⊆   .If  ∈   , then where then  ∈   such that  ≤ .This completes the proof.

𝐿−Fuzzy Prime Ideals
In this section we define −fuzzy prime ideals and investigate some of their properties.
Let  be a prime ideal of  and  be a prime element in .Consider an −fuzzy subset   of  defined by If  is a nontrivial algebra such that  ∈ [, ] for all  ∈ , then it can be deduced from the above theorem that −fuzzy prime ideals exist in .
For a nontrivial algebra , to have an −fuzzy −system is a sufficient condition for  to possess −fuzzy prime ideals.

Maximal Fuzzy 𝐿−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.

Theorem 15 .
For each  ∈  and −fuzzy ideals  and  of  Proof.For each  ∈ , let us define two sets   and   as follows.Proof.For each  ∈ , let us take the set   as in Theorem 15 and define a set   as follows.
. A nonconstant −fuzzy ideal  of  is called an −fuzzy prime ideal if and only if Notation .For −fuzzy points   and   of , we denote [⟨  ⟩, ⟨  ⟩] by [  ,   ].Theorem 22.A nonconstant −fuzzy ideal  of  is −fuzzy prime if and only if for any −fuzzy points   and   of  [  ,   ] ≤  ℎ   ∈     ∈ .
(26)Proof.Suppose that  satisfies the condition:[  ,   ] ≤  ⇒ ℎ   ∈     ∈ (27)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 () = {1, }, where  is a prime element in  and the set  * = { ∈  : () = 1} is a prime ideal of .Proof.Suppose that  is a prime −fuzzy ideal.Clearly 1 ∈ (), and since  is nonconstant, there is some  ∈  such that () < 1.We show that () = () for all ,  ∈  −  * .Let ,  ∈  such that () < 1 and () < 1.Let us define −fuzzy subsets  and  of  as follows: 34)for all  ∈ .The above theorem confirms that −fuzzy prime ideals of  are only of the form   .This establishes a one-toone 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 .Let  be an ideal of  and  a prime element in .en  is a prime ideal if and only if   is an −fuzzy prime ideal.)forall  ∈ [, ].Since  is prime, by Theorem 23 there exists a prime element  < 1 in  such that () =  = () and () = 1 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.∈, then  is −fuzzy prime.It is natural to ask ourselves, does every algebra in K 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.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 F = { ∈ FI() :  ≤   () ≤ }.Clearly  ∈ F so that F 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 F. Now it is enough to show that  is prime.Suppose not.Then there exist −fuzzy ideals  and ] of  such that [, ]] ≤  but Since  ∈ [, ] and  is ∧−irreducible element in , we get [ 1 ,  2 ]() ≰ , so () ≰ , which is a contradiction.Therefore  is prime.