Application of L -fuzzy sets in m -ary semigroups

1 Introduction and preliminaries

Fuzzy set theory was developed by Zadeh [29] introducing the notion of a fuzzy subset μ of a non-empty set X as a function from X to [0 ;1]. It is the most appropriate approach to deal with uncertainties and it has been applied to many branches in mathematics and other applied areas. Goguen in [13] generalized the notion of a fuzzy subset of X to that of an L-fuzzy subset, namely a function from X to a lattice L. Later Rosenfeld considered the fuzzification of algebraic structures [27] which opened a new direction, new exploration, new path of thinking to mathematicians, engineers, computer scientists, and many other researchers. In 1982, Liu [20], introduced and studied the notion of a fuzzy ideal of a ring. In 1988, Zhang [28] studied prime L-fuzzy ideals in rings where L is completely distributive lattice. Since then several authors have obtained interesting results on L-fuzzy ideals of rings, L-fuzzy ideals of semirings and L-fuzzy modules [14, 30–32] etc. See [21] for a comprehensive survey of the literature on thesedevelopments.

The generalization of algebraic structures was in active research for a long time, it was first initiated by Kasner [17] in 1904. The notion of an n-ary group was introduced in 1928 by W. Dörnte [3] (under inspiration of Emmy Noether), but the important study of n-ary semigroups and n-ary groups was done by Dudek. For more details, the reader is referred to [4–11, 22] etc. Up till now, the theory of n-ary systems have many applications, for example, in the theory of automata. We know that n-ary semigroups have been applied in the theory of fuzzy sets and rough sets (see [1, 2]). The first fuzzification of an n-ary system was introduced by Dudek [12]. Moreover, as a generalization of Rosenfeld’s fuzzy groups, Davvaz and Dudek [2] discussed further fuzzy n-ary groups, and investigated their related properties. Other recent results on fuzzy n-ary semigroups were obtained by Zhan et al. [33, 34].

In this paper, applying the theory of L-fuzzy sets, we introduce the concept of an L-fuzzy ideal (L-fuzzy k-ideal) of an m-ary semigroup. Some properties of them are investigated and some structural theorems for L-fuzzy ideals (L-fuzzy k-ideals) of m-ary semigroups are proved. In this direction the concept of image and preimage of an L-fuzzy set under m-ary semigroup homomorphism are discussed. Also, the notions of normal L-fuzzy ideal (L-fuzzy k-ideal) and maximal L-fuzzy ideal (L-fuzzy k-ideal) of an m-ary semigroup are introduced and some properties of them are studied. Further, we introduce the notion of L-fuzzy congruence on an m-ary semigroup and make a study of L-fuzzy quotient m-ary semigroup using an L-fuzzy congruence, an L-fuzzy quotient m-ary semigroup induced by L-fuzzy ideals. We also investigate some properties of homomorphisms betweenthem.

Part of the results can be seen also as being obtained in the spirit of the transfer principle in fuzzy theory [18].

Recall first the basic terms and definitions from the hyperstructure theory and soft set theory.

2 Algebraic systems and m-ary structures

In this section we recall some known notions on what is meant by an algebraic system and m-ary structure.

Let H be a nonempty set and f : H × H → H be a mapping. Then f is called a binary (algebraic) operation on H. In general, a mapping f : H × H × . . . × H → H where H appears m times, is called an m-ary (algebraic) operation, and m is called the arity of this operation. The pair (H, f), where f is an m-ary operation defined on H, is called an m-ary groupoidor an m-ary system.

Let f be an m-ary operation on H and A₁, A₂, . . . , A _m nonempty subsets of H. We define A 2 , . . . , A m ) = x 2 , . . . , x m ) | x i ∈ A i , i = 1 , 2 , . . . , m } .

We shall use the following abbreviated notation: the sequence x _i , x_i+1, . . . , x _j will be denoted by x i j . For j < i, x i j is the empty symbol. In thisconvention, f (x₁, . . . , x _i , y_i+1, . . . , y _j , z_j+1, . . . , z _m )

will be written as f ( x 1 i , y i + 1 j , z j + 1 m ) . In the case when y_i+1 = . . . = y _j  = y, the last expression will be written in the form f ( x 1 i , y ( j - i ) , z j + 1 m ) .

Similarly, for nonempty subsets A₁, A₂, . . . , A _m of H we define f ( A 1 m ) = f ( A 1 , A 2 , . . . , A m ) = ∪ { f ( x 1 m ) | x i ∈ A i , i = 1 , . . . , m } . An m-ary operation f is called (i, j) - associative if f ( x 1 i - 1 , f ( x i m + i - 1 ) , x m + i 2 m - 1 ) = = f ( x 1 j - 1 , f ( x j m + j - 1 ) , x m + j 2 m - 1 ) , holds for fixed 1 ≤ i < j ≤ m and all x₁, . . . , x_2m-1 ∈ H.

Note that (i, k)-associativity follows from (i, j)- and (j, k)-associativities.

If the above condition is satisfied for all i, j ∈ {1, 2, . . . , m}, then we say that f is associative.

By an algebraic system (H, f₁, f₂, . . . , f _m ) or simply H is meant a set H closed under a collection of m _i -ary operation f _i and often also satisfying a fixed set of laws, for instant, the associative law.

A subset S of H constitues a subsystem iff S is closed under the same operations and satisfies the same fixed laws in H.

Let H be an algebraic system. A k-ideal, k = 1, 2, . . . , m relative to the m-ary operation is defined to be a subsystem I _k such that for any x₁, . . . , x _m  ∈ H, if x _k  ∈ I _k then f ( x 1 k - 1 , x k , x k + 1 m ) ⊆ I k . The k-ideal relative to f generated by an element a ∈ H (usually called a principal k-ideal) is denoted by ( a ) k = f ( H , H , . . . , a k , . . . , H ) ∪ { a } .

A subsystem I which is a k-ideal for each k = 1, . . . , m is simply called an ideal.

An m-ary groupoid (H, f) will be called an m-ary semigroup if and only if f is associative.

An m-ary semigroup (S, f) is called idempotent if f (x, . . . , x) = x for all x ∈ S. An m-ary semigroup (S, f) has zero element 0 if it satisfies: f ( x 1 i - 1 , 0 , x i + 1 m ) = 0 for all x₁, x₂, . . . , x _m  ∈ S and 1 ≤ i ≤ m [8].

3 L-fuzzy ideals in m-ary semigroups

Let L = (L, ≤ , ∧ , ∨) be a completely distributive lattice, which has the least and the greatest elements, say 0 and 1, respectively. Let X be a non-empty (usual) set. An L-fuzzy set in X is a map μ : X → L, and F ( X ) will denote the set of all L-fuzzy sets in X. If μ , ν ∈ F ( X ) , then μ ⊆ ν if and only if μ (x) ≤ ν (x) for all x ∈ X, and μ ⊂ ν if and only if μ ⊆ ν and μ ≠ ν. It is easily seen that F ( X ) = ( F ( X ) , ⊆ , ∧ , ∨ ) is a completely distributive lattice, which has the least and the greatest elements, say 0 ¯ and 1 ¯ , respectively in natural manner, where 0 ¯ ( x ) = 0 , 1 ¯ ( x ) = 1 for all x ∈ X.

Given any two sets X and X′, let μ ∈ F ( X ) and let h : X → X′ be any function. For y ∈ X′, we define ν ∈ F ( X ′ ) by ν ( y ) = { x ∈ h - 1 ( y ) sup μ ( x ) , if h - 1 ( y ) ≠ ∅ , 0 , otherwise ,

we call ν the image of μ under h, written h (μ). For any ν ∈ F ( h ( X ) ) , we define μ ∈ F ( X ) by μ (x) = ν (h (x)) for all x ∈ X, and we call μ the preimage of ν under h which is denoted by h^-1 (ν).

Definition 3.1. Let (S, f) be an m-ary semigroup. An L-fuzzy set μ ∈ F ( S ) on S is called an L-fuzzy k-ideal if μ ( f ( x 1 m ) ) ≥ μ ( x k )

for all x₁, x₂, . . . , x _m  ∈ S. If μ is an L-fuzzy k-ideal for every k = 1, 2, . . . , m, then it is called an L-fuzzy ideal of S. Clearly, if μ satisfies μ ( f ( x 1 m ) ) ≥ μ ( x 1 ) ∨ μ ( x 2 ) ∨ . . . ∨ μ ( x m ) , then it is an L-fuzzy ideal of S.

It follows that for an m-ary semigroup (S, f) with zero, if μ is an L-fuzzy k-ideal of S, then μ (0) ≥ μ (x) for all x ∈ S.

Example 3.2. [5] Let ℕ be the set of all natural numbers with zero included, S = ℕ and f an m-ary operation (m > 2) defined on S by the formula f ( x 1 m ) = x 1 · x 2 · . . . · x m . Then (S, f) is an m-ary semigroup with zero element 0. Define an L-fuzzy set of (S, f) by μ ( x ) = { s , if x is an even number , t , otherwise

where s, t ∈ L such that 0 ≤ t ≤ s ≤ 1. Then μ is an L-fuzzy ideal of S.

Example 3.3. Let S = ℕ be the set of all natural numbers with zero included and f an m-ary operation (m > 2) defined on S by the formula f ( x 1 m ) = x 1 · x 2 · . . . · x m . Then (S, f) is an m-ary semigroup. Let L = {0, α, β, 1} be the Boolean algebra of four elements. Define an L-fuzzy set of (S, f) by μ ( x ) = { 1 , if x∈ 6 ℕ , α , if x∈ 2 ℕ\ 6 ℕ , β , if x∈ 3 ℕ\ 6 ℕ , 0 , everywhereelse

Then μ is an L-fuzzy ideal of S.

Example 3.4. Let S = {- i, 0, i} be a set with a 3-ary operation f as the usual multiplication of complex numbers. Then (S, f) is a 3-ary semigroup. Define an L-fuzzy set μ : R → L by μ (- i) = μ (i) < μ (0). Then μ is an L-fuzzy ideal of S.

Example 3.5. Let S = {(0, 0) , (0, 1) , (1, 0) , (1, 1)}. Then S is a 3-ary semigroup with respect to ternary multiplication defined by (i, j) (k, l) (m, n) = (i, n). Let A = {(0, 0) , (0, 1)} be a subset of S. Define an L-fuzzy set μ in S as follows: μ (x) = α if x ∈ A, where α ∈ L such that α > 0 and μ (x) =0 otherwise. Then μ is an L-fuzzy k-ideal for some k = 1, 2, 3 but it is not an L-fuzzy ideal.

Let (S, f) be an m-ary semigroup, μ ∈ F ( S ) and t ∈ L. Let μ _t  = {x ∈ S|μ (x) ≥ t}, which is called a level subset of μ.

Remark 3.6. Note that

for s, t ∈ L, s < t implies μ _t  ⊆ μ _s and

Proposition 3.7. Let (S, f) be an m-ary semigroup and let μ ∈ F ( S ) . μ is an L-fuzzy k-ideal (k = 1, 2, . . . , m) (ideal) of S if and only if, for any t ∈ L such that μ _t ≠ ∅, μ _t is an k-ideal (ideal) of S.

Proof. Let μ be an L-fuzzy k-ideal of S and let t ∈ L such that μ _t ≠ ∅. Let x₁, . . . , x _m  ∈ S and x _k  ∈ μ _t . We have μ (f (x₁, . . . , x _m )) ≥ μ (x _k ) ≥ t. Thus f (x₁, . . . , x _m ) ∈ μ _t . Therefore μ _t is an k-ideal of S. Conversely, let all μ _t ≠ ∅ be k-ideal of S. Let x₁, . . . , x _m  ∈ S and t = min {μ (x₁) , . . . , μ (x _m )}. Then μ (x₁) , . . . , μ (x _m ) ≥ t, thus x₁, . . . , x _m  ∈ μ _t . Let s = μ (x _k ). Then μ (x _k ) ≥ s. Thus x _k  ∈ μ _s . By assumption, we have f ( x 1 k - 1 , x k , x k + 1 m ) ∈ μ s for all x₁, . . . , x _m  ∈ S. It follows that μ ( f ( x 1 k - 1 , x k , x k + 1 m ) ) ≥ s = μ ( x k ) . Therefore μ is an L-fuzzy k-ideal of S.

Corollary 3.8. Let (S, f) be an m-ary semigroup with zero. If μ ∈ F ( S ) is an L-fuzzy k-ideal (ideal) of S, then the set S _μ  = {x ∈ S|μ (x) ≥ μ (0)} is an k-ideal (ideal) of S.

If μ is an L-fuzzy k-ideal (ideal) of S, we call μ _t  (≠ ∅) a level k-ideal (ideal) of μ.

Theorem 3.9. Let (S, f) be an m-ary semigroup. If A is an k-ideal (ideal) of S, then there exists an L-fuzzy k-ideal (ideal) μ of S such that μ _t  = A for some t ∈ L.

Proof. Let t ∈ L and we define an L-fuzzy set of S by μ ( x ) = { t , if x ∈ A 0 , otherwise ,

then it follows that μ _t  = A. For s ∈ L, we have μ s = { μ 0 ( = S ) , if s = 0 μ t ( = A ) , if s ≤ t ∅ , if t < s ≤ 1 .

Since S and A are k-ideals of S, it follows that every non-empty level subset μ _s of μ is a k-ideal of S. By Proposition 3.7, μ is an L-fuzzy k-ideal of S, which completes the proof.

Theorem 3.10. Let (S, f) be an m-ary semigroup and let μ ∈ F ( S ) be an L-fuzzy k-ideal of S. Then two level k-ideals μ _s , μ _t of μ, where s < t in L, are equal if and only if there is no x ∈ S such that s ≤ μ (x) < t.

Proof. Let s, t ∈ L such that s < t and μ _s  = μ _t . If there exists x ∈ S such that s ≤ μ (x) < t, then x ∈ μ _s but x ∉ μ _t , which is impossible. Conversely, suppose that there is no x ∈ S such that s ≤ μ (x) < t. But s < t implies μ _t  ⊆ μ _s . If y ∈ μ _s , then μ (y) ≥ s, and so μ (y) ≥ t because μ (y) notlessthant. Hence y ∈ μ _t , and this implies that μ _s  = μ _t . This completes theproof.

Theorem 3.11. Let (S, f) be an m-ary semigroup. The following statements hold:

Let μ ∈ F ( S ) be an L-fuzzy subset of S. If Im (μ) = {t₁, t₂, . . . , t _n }, where t₁ < t₂ < . . . < t _n , then the family of sets μ _{t i}  (i = 1, . . . , n) constitutes the collection of all level subsets of μ.

Proof. (1) If t ∈ L with t < t₁, then μ _t  = μ _{t 1} . If t ∈ L with t > t _n , we have that μ _t  =∅. If t ∈ L with t _i  < t < t_i+1 for some i = 1, 2, . . . , n - 1, by assumption, there is no x ∈ S such that t ≤ μ (x) < t_i+1. By Theorem 3.10, μ _t  = μ _{t i+1} .

(2) Let us suppose that t ∈ L with t < t₁, then μ _{t 1}  ⊆ μ _t . Since μ _{t 1}  = S, we have μ _t  = S and μ _t  = μ _{t 1} . Let now suppose that t ∈ L with t _i  < t < t _i  + 1 (1 ≤ i ≤ n - 1), then there is no x ∈ S such that t ≤ μ (x) < t _i  + 1. It follows from Theorem 3.10 that μ _t  = μ _{t i+1} . Thus we have that for any t ∈ L with t ≤ μ (0), the level k-ideal μ _t is in {μ _{t i} |1 ≤ i ≤ n}. This completes the proof.

Theorem 3.12. Let (S, f) , (R, g) be two m-ary semigroups and h : S → R be an onto homomorphism. Then the preimage of an L-fuzzy k-ideal (ideal) ν under h is an L-fuzzy k-ideal (ideal) of S.

Proof. Let h : S → R be an onto homomorphism. Let ν ∈ F ( R ) be an L-fuzzy k-ideal and let μ be the preimage of ν under h. Then for any x₁, x₂, . . . . , x _m  ∈ S, we have for k ∈ {1, 2, . . . , m}: μ (f (x₁, . . . . , x _m )) = ν (h (f (x₁, . . . . , x _m )) = ν (g (h (x₁) , . . . . , h (x _m )) ≥ ν (h (x _k )) = μ (x _k ).

This shows that μ is an L-fuzzy k-ideal of S.

Proposition 3.13. Let h be a mapping from a set X to a set Y, and let μ ∈ F ( X ) . Then for every t ∈ L, t ≠ 0, (h (μ))  _t  = underset0 < s < t ⋂ h (μ_t-s).

Proof. Let t ∈ L, t ≠ 0. If y ∈ (h (μ))  _t , then t ≤ (h (μ)) (y) = undersetz ∈ h^-1 (y) sup μ (z). It follows that there exists x₀ ∈ h^-1 (y), such that μ (x₀) > t - s for all s ∈ L, where 0 < s < t, and so y = h (x₀) ∈ h (μ_t-s). Therefore y ∈ underset0 < s < th (μ_t-s). Conversely, let y ∈ underset0 < s < t ⋂ h (μ_t-s). Then y ∈ h (μ_t-s) for all s ∈ L, where 0 < s < t. This implies that there exists x₀ ∈ μ_t-s such that y = h (x₀). It follows that h (x₀) ≥ t - s and x₀ ∈ h^-1 (y), so that (h (μ)) (y) = undersetz ∈ h^-1 (y) μ (z) ≥ underset0 < s < t sup {t - s} = t. Therefore y ∈ (h (μ))  _t . This completes the proof.

Theorem 3.14. Let (S, f) , (R, g) be two m-ary semigroups. Let h : S → R be an onto homomorphism. If μ is an L-fuzzy k-ideal (ideal) of S, then the homomorphic image h (μ) is an L-fuzzy k-ideal (ideal) of R.

Proof. By Proposition 3.7 it is sufficient to show that each non-empty level subset of h (μ) is an k-ideal of R. Let (h (μ))  _t be a non-empty level subset of h (μ) for all t ∈ L. If t = 0, then (h (μ))  _t  = R. Assume that t ≠ 0. By Proposition 3.13, (h (μ))  _t  = underset0 < s < t ⋂ h (μ _s ). Then h (μ _s ) is non-empty for all 0 < s < t, and so μ _s is a non-empty level subset of μ for all 0 < s < t. Since μ is an L-fuzzy k-ideal of S, it follows from Proposition 3.7 that μ _s is an k-ideal of S. Since h is an onto homomorphism, h (μ _s ) is an k-ideal of R. Hence (h (μ))  _t being an intersection of a family of k-ideals is also an k-ideals of R. This completes theproof.

Definition 3.15. Let (S, f) be an m-ary semigroup. A k-ideal A of S is said to be characteristic if h (A) = A for all h ∈ Aut (S), where Aut (S) is the set of all automorphisms of S. An L-fuzzy k-ideal μ of S is said to be L-fuzzy characteristic if μ (h (x)) = μ (x) for all x ∈ S and h ∈ Aut (S).

Theorem 3.16. Let (S, f) be an m-ary semigroup. If μ is an L-fuzzy k-ideal (ideal) of S and h : S → S is an onto homomorphism, then the mapping μ h ∈ F ( S ) , defined by μ ^h  (x) = μ (h (x)) for all x ∈ S, is an L-fuzzy k-ideal (ideal) of S.

Proof. For any x₁, . . . , x _m  ∈ S, we have μ h ( f ( x 1 , . . . , x m ) ) = μ ( h ( f ( x 1 , . . . , x m ) ) ) = μ ( f ( h ( x 1 ) , . . . , h ( x m ) ) ) ≥ μ ( h ( x k ) ) = μ h ( x k ) , where k ∈ {1, 2, . . . , m}. Therefore μ ^h is an L-fuzzy k-ideal of S.

Theorem 3.17. Let (S, f) be an m-ary semigroup. Let μ be an L-fuzzy k-ideal (ideal) of S. Then μ is an L-fuzzy characteristic k-ideal (ideal) of S if and only if each level k-ideal (ideal) of μ is characteristic.

Proof. Let μ be an L-fuzzy characteristic k-ideal of S and let h ∈ Aut (S). For all t ∈ L, if y ∈ h (μ _t ), then μ (y) = μ (h (x)) = μ (x) ≥ t for some x ∈ μ _t with y = h (x). It follows that y ∈ μ _t . Conversely, if y ∈ μ _t , then t ≤ μ (y) = μ (h (x)) = μ (x) for some x ∈ S, where y = h (x). It follows that y ∈ h (μ _t ).

Let we assume now that each level k-ideal of μ is characteristic. Let x ∈ S and h ∈ Aut (S). If μ (x) = t ∈ L, then by Remark 3.6, x ∈ μ _t and x ∉ μ _s for all s > t. Since each level k-ideal of μ is characteristic, then h (x) ∈ h (μ _t ) = μ _t . Assume μ (h (x)) = s > t. Then h (x) ∈ μ _s  = h (μ _s ). Since h is one-to-one, then it follows that x ∈ μ _s , which is impossible. Therefore μ (h (x)) = t = μ (x), that is, μ is L-fuzzy characteristic.

Lemma 3.18. Let (S, f) be an m-ary semigroup and μ be an L-fuzzy k-ideal of S. If Im (μ) is finite, that is Im (μ) = {t₁, . . . , t _n }, then for all t _i , t _j  ∈ Im (μ) , μ _{t i}  = μ _{t j} implies t _i  = t _j .

Proof. Let us suppose that t _i  ≠ t _j , assume t _i  < t _j . Since t _i  ∈ Im (μ), there is an x ∈ S such that μ (x) = t _i and hence t _i  ≤ μ (x) < t _j . By Theorem 3.10, μ _{t i}  = μ _{t j} does not hold, which is impossible.

Theorem 3.19. Let (S, f) be an m-ary semigroup and let μ and ν be two L-fuzzy k-ideals of S with a single family of level k-ideals. If Im (μ) = {t₀, t₁, . . . , t _r } and Im (ν) = {s₀, s₁, . . . , s _k } where t₀ > t₁ > . . . > t _r and s₀ > s₁ > . . . > s _k , then

r = k,

Proof. By Theorem 3.10, since μ and ν have the same family of level k-ideals, it follows that r = k. By Theorem 3.10, we have two chains of level k-ideals μ _{t 0}  ⊂ μ _{t 1}  ⊂ . . . ⊂ μ _{t k}  = S and ν _{s 0}  ⊂ ν _{s 1}  ⊂ . . . ⊂ ν _{s k}  = S.

It follows that if t _i , t _j  ∈ Im (μ), where t _i  > t _j , then μ _{t i}  ⊂ μ _{t j} (*). If s _i , s _j  ∈ Im (ν), where s _i  > s _j , then ν _{s i}  ⊂ ν _{s j} (**). Since the two families of level k-ideals are identical, it is clear that μ _{t 0}  = ν _{s 0} . We have μ _{t 1}  = ν _{t j} since j > 0. Assume that μ _{t 1}  ≠ ν _{t 1} . Then μ _{t 1}  = ν _{s j} for some j > 1, and ν _{s 1}  = μ _{t i} for some t _i  < t₁. Thus by (*) and (**), we obtain ν _{s j}  = μ _{t 1}  ⊂ μ _{t i} and μ _{t i}  = ν _{s 1}  ⊂ ν _{s j} , which is impossible. Therefore μ _{t 1}  = ν _{s 1} . By induction on i, we obtain that μ _{t i}  = ν _{s i}  (0 ≤ i ≤ k). Let x ∈ S, where μ (x) = t _i and ν (x) = s _j for some i, j ∈ {0, 1, . . . , k}. We have s _i  = s _j . Indeed: if μ (x) = t _i , then by (2), x ∈ μ _{t i}  = ν _{s i} . It follows that s _j  = ν (x) > s _i and from (**), we have ν _{s j}  ⊆ ν _{s i} . Now, by (2), ν (x) = s _j implies x ∈ ν _{s j}  = μ _{t j} . It follows from (*) that t _j  ≤ μ (x) = t _i and μ _{t 1}  ⊆ μ _{t j} . Therefore we have ν _{s i}  = μ _{t i}  ⊆ μ _{t j}  = ν _{s j} . Thus ν _{s i}  = ν _{s j} , and by Lemma 3.18, we have s _i  = s _j . This completes the proof.

Theorem 3.20. Let (S, f) be an m-ary semigroup and μ, ν be two L-fuzzy k-ideals of S having the same family of level k-ideals. Then μ = ν if and only if Im (μ) = Im (ν).

Proof. Let us suppose that Im (μ) = Im (ν). Assume that Im (μ) = {t₀, t₁, . . . , t _r } and Im (ν) = {s₀, s₁, . . . , s _r }, where t₀ > t₁ > . . . > t _r and s₀ > s₁ > . . . > s _r . Then s₀ ∈ Im (ν) = Im (μ) and thus s₀ = t _{k 0} for some k₀. Assume that t _{k 0}  ≠ t₀. So t _{k 0}  < t₀. Now, s₁ ∈ Im (ν) = Im (μ), and hence s₁ = t _{k 1} for some k₁. Since s₀ > s₁, we have t _{k 0}  > t _{k 1} . Continuing in this way, we have t₀ > t _{k 0}  > t _{k 1}  > . . . > t _{k r} . This means that |Im (μ) | = r + 1. Hence we have s₀ = t₀. Proceeding in this manner, we obtain that s _i  = t _i , (0 ≤ i ≤ r). Let x ∈ S, with μ (x) = t _i for some i ∈ {0, 1, . . . , r}. Then by Theorem 3.19, ν (x) = s _i  (0 ≤ i ≤ r). Since s _i  = t _i , it follows that μ (x) = ν (x) for each x ∈ S. Hence μ = ν. This completes the proof.

Definition 3.21. Let (S, f) be an m-ary semigroup. An L-fuzzy subset μ of S is said to be normal if μ (0) =1.

Let (S, f) be an m-ary semigroup and μ ∈ F ( S ) . Define an L-fuzzy subset μ⁺ of S by μ⁺ (x) = μ (x) ∨ (μ (0)) ′, for all x ∈ S, where L is a complemented distributive lattice and (μ (0)) ′ is the complement of μ (0).

Proposition 3.22. Let (S, f) be an m-ary semigroup and μ ∈ F ( S ) . The following statements hold:

μ⁺ is a normal L-fuzzy subset of S containing μ.

Proof. (1) We can see that μ⁺ (0) = μ (0) ∨ (μ (0)) ′ = 1 and for x ∈ S, μ (x) ≤ μ⁺ (x). This completes the proof.

(2) By (1), we have (μ⁺) ⁺ (x) = μ⁺ (x) ∨ (μ⁺ (0)) ′ = μ⁺ (x) ∨1′ = μ⁺ (x). This completes the proof.

(3) Assume that μ is normal. Then μ⁺ (x) = μ (x) ∨ (μ (0)) ′ = μ (x) ∨1′ = μ (x). The converse is obvious by (1).

Corollary 3.23. Let (S, f) be an m-ary semigroup, μ ∈ F ( S ) and x ∈ S. If μ⁺ (x) =0, then μ (x) =0.

Proof. By Proposition 3.22(1), we have μ (x) < μ⁺ (x). it follows that μ (x) =0.

Theorem 3.24. Let (S, f) be an m-ary semigroup and μ ∈ F ( S ) . If μ is an L-fuzzy k-ideal (ideal) of S, then μ⁺ is a normal L-fuzzy k-ideal (ideal) of Scontaining μ.

Proof. Let x₁, . . . , x _m  ∈ S. Then μ + ( f ( x 1 , . . . , x m ) ) = μ ( f ( x 1 , . . . , x m ) ) ∨ ( μ ( 0 ) ) ′ ≥ μ ( x k ) ∨ ( μ ( 0 ) ) ′ ≥ μ + ( x k ) Therefore μ⁺ is an L-fuzzy k-ideal of S. By Proposition 3.22(1), μ⁺ is a normal L-fuzzy k-ideal of S containing μ.

Theorem 3.25. Let (S, f) be an m-ary semigroup and μ ∈ F ( S ) . If μ ∈ F ( S ) is an L-fuzzy k-ideal (ideal) of S and if there exists ν an L-fuzzy k-ideal (ideal) of S satisfying ν⁺ ⊂ μ, then μ⁺ is a normal L-fuzzy k-ideal (ideal) of S.

Proof. Let us suppose that there exists ν an L-fuzzy k-ideal of S satisfying ν⁺ ⊂ μ. Then we have 1 = ν⁺ (0) ≤ μ (0). It follows that μ (0) =1. This completes the proof.

By Proposition 3.22 and Theorem 3.25 we have the following:

Corollary 3.26. Let (S, f) be an m-ary semigroup, μ ∈ F ( S ) be an L-fuzzy k-ideal of S. If μ ∈ F ( S ) is an L-fuzzy k-ideal (ideal) of S and if there exists ν an L-fuzzy k-ideal (ideal) of S satisfying ν⁺ ⊂ μ, then μ⁺ = μ.

Let (S, f) be an m-ary semigroup and (N (S) , ⊆) denotes the partially ordered set of normal L-fuzzy ideals of S under set inclusion. A non constant L-fuzzy ideal μ of S is said to be a maximal L-fuzzy ideal if μ⁺ is a maximal element of (N (S) , ⊆).

Theorem 3.27. Let (S, f) be an m-ary semigroup. If μ is a non constant maximal normal L-fuzzy ideal of S, then μ takes the values only 0 and 1.

Proof. Let y ∈ S, μ be a maximal normal L-fuzzy ideal of S, 0 < μ (y) <1 and μ (y) = a. Define L-fuzzy subset ν of S by ν (x) = μ (x) ∨ a, for all x ∈ S. Then ν (x) ≥ μ (x) for all x ∈ S. ν is a normal L-fuzzy ideal of S, since ν (0) = μ (0) ∨ a = 1 ∨ a = 1. If x ≠ 0, μ (x) < ν (x). Therefore μ is not a maximal, which is impossible. This completes the proof.

Theorem 3.28. Let (S, f) be an m-ary semigroup. If μ is a maximal L-fuzzy ideal of S, then S _μ is a maximal ideal of S.

Proof. Let μ be a maximal L-fuzzy ideal of S. Then μ⁺ is a maximal element of (N (S) , ⊆). By Theorem 3.27, μ⁺ takes only the values 0 and 1. If μ⁺ (x) =1, then μ (x) ∨ (μ (0)) ′ = 1. It follows that μ (0) =1, since μ (0) ≥ μ (x), for all x ∈ S. We have μ (x) ≤ μ⁺ (x), for all x ∈ S. If μ⁺ (x) =0, then μ (x) ∨ (μ (0)) ′ = 0. It follows that μ (x) =0 and (μ (0)) ′ = 0, which implies that μ (0) =1. Therefore μ is a normal L-fuzzy ideal of S. S _μ is a proper ideal of S, since μ is a non constant. Let A be an ideal of S such that S _μ  ⊆ A. It follows that χ _{S μ}  ⊆ χ _A which implies that μ = χ _{S μ}  ⊆ χ _A . Since μ and χ _A are normal L-fuzzy ideals of S and μ = μ⁺. It follows that μ is a maximal element of N (S) which implies that μ = χ _A or χ _A  = 1. If chi _A  = 1, then A = S. If μ = χ _A , then S _μ  = S _{χ A}  = A. Therefore S _μ is a maximal ideal of S.

4 L-fuzzy congruence relation on m-ary semigroup

In this section we introduce the notion of L-fuzzy congruence on an m-ary semigroup (S, f) and make a study of L-fuzzy quotient m-ary semigroup using an L-fuzzy congruence, an L-fuzzy quotient m-ary semigroup induced by L-fuzzy ideals. We also investigate some properties of homomorphisms between them. These results can be seen as an extension and generalization of those obtained in [16], [19], [24].

Let (S, f) be an m-ary semigroup. Let R be an equivalence relation of (S, f). R is called a congruence relation of (S, f) if it satisfies: (x _i , y _i ) ∈ R implies ( f ( x 1 m ) , f ( y 1 m ) ) ∈ R

for all 1 ≤ i ≤ m, x₁, . . . , x _m , y₁, . . . , y _m  ∈ S.

Let X and Y be two non-empty sets. Zadeh [29] introduced the definition of a fuzzy relation from X to Y as a fuzzy subset of X × Y. An L-fuzzy relation from X to Y is defined to be an L-fuzzy subset of X × Y. If R is an L-fuzzy relation from X to Y, we denote R ∈ F ( X × Y ) . Let R ∈ F ( X × Y ) and Q ∈ F ( X × Y ) . The max-min composition of R and Q is defined as a L-fuzzy relation Q ∘ R from X to Z such that (Q ∘ R) (x, z) = ⋁  _y R (x, y) ∧ Q (y, z), for all (x, z) ∈ X × Z.

Let (S, f) be an m-ary semigroup and R ∈ F ( S × S ) . Then R is called an L-fuzzy equivalence relation on S iff

R is reflexive i.e., R (x, x) =1, for all x ∈ S.

Let R be an L-fuzzy equivalence relation on m-ary semigroup (S, f). Then for each α ∈ L, we define two crisp relations on S as follows:

Definition 4.1. A weak α-relation denoted by ω _α is defined on S as xω _α y if and only if R (x, y) ≥ α and α-strong relation denoted by σ _α is defined on S as xσ _α y if and only if R (x, y) > α, for any x, y ∈ S. We call the L-fuzzy quotient set, the set S/R = {[a]  _R  : a ∈ S}, where R is an L-fuzzy equivalence relation on S.

The following statement holds:

Lemma 4.2. Let R be an L-fuzzy equivalence relation on a set X. Then

R (a, b) =0 if and only if [a]  _R  ∧ [b]  _R  ≡ 0,

Definition 4.3. An L-fuzzy set δ is an L-fuzzy m-ary semigroup in S if for x₁, . . . , x _m  ∈ S, δ (f (x₁, . . . , x _m )) ≥ δ (x₁) ∧ . . . ∧ δ (x _m ).

Note that the ∩ of any set of L-fuzzy semigroups is an L-fuzzy semigroup.

We define now an L-fuzzy congruence relation on an m-ary semigroup (S, f).

Definition 4.4. An L-fuzzy relation R on S is said to be compatible if R ( f ( x 1 m ) , f ( y 1 m ) ) ≥ R ( x i , y i ) , for all 1 ≤ i ≤ m, x₁, . . . , x _m , y₁, . . . , y _m  ∈ S. A compatible L-fuzzy equivalence relation on S is called anL-fuzzy congruence.

Note that the ∩ of any L-fuzzy congruence relations on S is an L-fuzzy congruence relation.

Theorem 4.5. Let (S, f) be an m-ary semigroup. If R is an L-fuzzy congruence relation on S, then the following statements hold:

for all α ∈ L, ω _α is a crisp congruence relation on S

Proof. (1) Let α ∈ L. For every x ∈ S, R (x, x) =1 ≥ α, that is, xω _α x. For x, y ∈ S, xω _α y implies that R (x, y) ≥ α. Thus R (y, x) = R (x, y) ≥ α which implies that yω _α x. Let assume now that xω _α y and yω _α z. Then R (x, z) = (R∘ R) (x, z) = ⋁ _a∈S (R (x, a) ∧ R (a, z)) ≥ R (x, y) ∧ R (y, z) ≥ α ∧α = α. Therefore ω _α is an equivalence relation of S. Let now x _i ω _α y _i . It follows that R (x _i , y _i ) ≥ α. Thus R ( ( f ( x 1 m ) , f ( y 1 m ) ) ≥ R ( x i , y i ) ≥ α , for all 1 ≤ i ≤ m, x₁, . . . , x _m , y₁, . . . , y _m  ∈ S. This implies f ( x 1 m ) ω α f ( y 1 m ) , for all 1 ≤ i ≤ m, x₁, . . . , x _m , y₁, . . . , y _m  ∈ S. Therefore ω _α is a congruence relation on S.

(2) The proof is similar to the proof of (1).

Theorem 4.6. Let (S, f) be an m-ary semigroup and let R be an L-fuzzy congruence on S. Then S/R is an m-ary semigroup under the m-ary operation g defined by g ([x₁]  _R , . . . . , [x _m ]  _R ) = [f (x₁, . . . , x _m )]  _R for any x₁, . . . , x _m  ∈ S and ρ : S → S/R : x ⟼ [x]  _R is a homomorhism.

Proof. By Lemma 4.2(3), [a]  _R  = [b]  _R if and only if R (a, b) =1. Let x₁, . . . , x _m , y₁, . . . , y _m  ∈ S such that [x₁]  _R  = [y₁]  _R , . . . , [x _m ]  _R  = [y _m ]  _R . Then R (x₁, y₁) =1, . . . . , R (x _m , y _m ) =1. We have R (f (x₁, . . . , x _m ) , f (y₁, . . . , x _i , . . . , y _m )) ≥ R (x _i , y _i ) =1 and R (f (x₁, . . . , y _i , . . . , x _m ) , f (y₁, . . . , y _m )) ≥ R (x _i , y _i ) =1, for all 1 ≤ i ≤ m, x₁, . . . , x _m , y₁, . . . , y _m  ∈ S. Since R ( f ( x 1 , ... , x m ) , f ( y 1 , ... , y m ) ) ≥ ( R ∘ R ) ( f ( x 1 , ... , x m ) , f ( y 1 , ... , y m ) ) = ⋁ z ∈ S ( R ( f ( x 1 , ... , x m ) , z ) ∧ R ( z , f ( y 1 , ... , y m ) ) ) = 1 , therefore [f (x₁, . . . , x _m )]  _R  = [f (y₁, . . . , y _m )]  _R . It is clear that this m-ary operation g on S/R is associative. Furthermore, there exists the surjection ρ : S → S/R : x ⟼ [x]  _R . Since g (ρ (x₁) , . . . , ρ (x _m )) = g ([x₁]  _R , . . . , [x _m ]  _R ) = [f (x₁, . . . , x _m )]  _R   =  ρ (f (x₁, . . . , x _m )), ρ is homomorphism.

Theorem 4.7. Let (S, f) be an m-ary semigroup and μ an L-fuzzy set of S, and R^* (x, y) = (μ (x) ∧ μ (y)) ∨ Id _S  (x, y) for all x, y ∈ S. Then R^* is an L-fuzzy congruence on S.

Proof. It can be easily seen that R^* is symmetric and reflexive. For x _i , y _i  ∈ S we have R * ) ( x i , y i ) = ⋁ z ∈ S ( R * ( x i , z ) ∧ R * ( z , y i ) ) ⋁ z ∈ S ( { ( μ ( x i ) ∧ μ ( z ) ) ∨ Id S ( x i , z ) } { ( μ ( z ) ∧ μ ( y i ) ) ∨ Id S ( z , y i ) } ) { μ ( x i ) ∧ μ ( y i ) } = R * ( x i , y i ) . Thus R is transitive. Therefore R is an L-fuzzy equivalence relation of S.

Let we show now the compatibility of R. We have R^* (f (x₁, . . . , x _m ) , f (y₁, . . . , y _m )) = (μ (f (x₁, . . . , x _m )) ∧ μ (f (y₁, . . . , y _m ))) ∨ Id _S  (f (x₁, . . . , x _m ) , f(y₁, . . . , y _m )), for all 1 ≤ i ≤ m, x₁, . . . , x _m , y₁, . . . , y _m  ∈ S. Then we have R * ( f ( x 1 , ... , x m ) , f ( y 1 , ... , y m ) ) = ( μ ( f ( x 1 , ... , x m ) ) ∧ μ ( f ( y 1 , ... , y m ) ) ) ∨ I d S ( f ( x 1 , ... , x m ) , f ( y 1 , ... , y m ) ) ≥ ( μ ( x i ) ∧ μ ( y i ) ) = R ( f ( x 1 , ... , x m ) , f ( y 1 , ... , y m ) ) . Therefore R is an L-fuzzy congruence on S.

Theorem 4.8. Let (S, f) and (S′, h) be two m-ary semigroups and let p : S → S′ be an m-ary semigroup (onto) homomorphism. Let μ, ν be L-fuzzy ideals of S and S′, respectively, such that μ ^p  ⊂ ν. Then there is a homomorphism of m-ary semigroups p^* : S/μ^* → S/ν^*, where μ^* (x, y) = (μ (x) ∧ μ (y)) ∨ Id _S  (x, y) and ν^* (x, y) = (ν (x) ∧ ν (y)) ∨ Id _S  (x, y), for all x, y ∈ S.

Proof. By Theorem 4.7, it follows that μ^* and ν^* are L-fuzzy congruences on S and S′ respectively, and thus [x]  _{μ ^*}  = [y]  _{μ ^*} if and only if μ^* (x, y) =1. Let us assume that [x]  _{μ ^*}  = [y]  _{μ ^*} for x, y ∈ S. Since μ^* (x, y) =1, then μ (x) ∧ μ (y) =1. Thus ν (p (x)) ∧ ν (p (y)) ≥ μ (x) ∧ μ (y) =1 and so ν * (p (x) , p (y)) =1. Therefore p^* is well-defined. It can be easily seen that p^* is a homomorphism.

Theorem 4.9. Let (S, f) and (S′, h) be two m-ary semigroups and let p : S → S′ be an m-ary semigroup homomorphism. Let μ, ν be L-fuzzy ideals of S and S′, respectively such that ν ^{p -1}  = μ. If we define μ^** (x, y) = (μ (x) ∧ μ (y)) ∨ Id_S′ (p (x) , p (y)), then μ^** is an L-fuzzy congruence on S.

Proof. It can be easily seen that μ^** is symmetric and reflexive. For x _i , y _i  ∈ S we have ( μ * * ∘ μ * * ) ( x i , y i ) = ⋁ z ∈ S ( μ * * ( x i , z ) ∧ μ * * ( z , y i ) ) = ⋁ z ∈ S ( { ( μ ( x i ) ∧ μ ( z ) ) ∨ I d S ' ( p ( x i ) , p ( z ) ) } ∧ { ( μ ( z ) ∧ μ ( y i ) ) ∨ I d S ' ( p ( z ) , p ( y i ) ) } ) ≥ ( μ ( x i ) ∧ μ ( z ) ) ∨ I d S ' ( p ( x i ) , p ( z ) ) = μ * * ( x i , y i ) . Thus μ^** is transitive. Therefore μ^** is an L-fuzzy equivalence relation of S. Let we show now the compatibility of μ^**. We have for all 1 ≤ i ≤ m, x₁, . . . , x _m , y₁, . . . , y _m  ∈ S, μ * * ( f ( x 1 , . . . , x m ) , f ( y 1 , . . . , y m ) ) = ( μ ( f ( x 1 , . . . , x m ) ) ∧ μ ( f ( y 1 , . . . , y m ) ) ) ∨ Id S ′ ( f ( x 1 , . . . , x m ) , f ( y 1 , . . . , y m ) ) ≥ ( μ ( x i ) ∧ μ ( y i ) ) ∨ Id S ′ ( f ( x 1 , . . . , x m ) , f ( y 1 , . . . , y m ) ) ≥ μ * * ( x i , y i ) . Therefore μ^** is an L-fuzzy congruence on S.

Theorem 4.10. Let (S, f) and (S′, h) be two m-ary semigroups and let p : S → S′ be an m-ary semigroup onto homomorphism. Let μ, ν be L-fuzzy ideals of S and S′, respectively such that ν ^{p -1}  = μ. Then S/μ^** ≅ S/ν^*.

Proof. It can be seen that p^** : S/μ^** → S/ν^* is onto homomorphism in similar way as in Theorem 4.8. Let we show that p^** is injective. Let y₁, y₂ ∈ S′ and let us suppose that [y₁]  _{ν ^**}  = [y₂]  _{ν ^**} . Then there exist x₁, x₂ ∈ S such that p (x₁) = y₁ and p (x₂) = y₂, since p is onto homomorphism. If y₁ = y₂, then μ^** (x₁, x₂) =1 and hence [x₁]  _{μ ^**}  = [x₂]  _{μ ^**} . If y₁ ≠ y₂, then we have ( μ ( x 1 ) ∧ μ ( x 2 ) ) ∨ Id S ′ ( p ( x 1 ) , p ( x 2 ) ) μ ( x 1 ) ∧ μ ( x 2 ) = ν ( p ( x 1 ) ) ∧ ν ( p ( x 2 ) ) ν * ( y 1 , y 2 ) = 1 . This completes the proof.

5 Conclusions

In this paper, we introduced the concept of an L-fuzzy ideal (L-fuzzy k-ideal) of an m-ary semigroup and some properties of them were studied. Further, we introduced the notion of L-fuzzy congruence on an m-ary semigroup (S, f) and made a study of L-fuzzy quotient m-ary semigroup using an L-fuzzy congruence, an L-fuzzy quotient m-ary semigroup induced by L-fuzzy ideals. Some properties of homomorphisms between them were investigated. These results will serve as a base for future work, combining L-fuzzy sets, soft sets, rough sets and applying them in real-life applications inspired by [35–39].