On almost (m, n)-ideals and fuzzy almost (m, n)-ideals in semigroups

In this paper, we define almost -ideals of semigroups by using the concepts of -ideals and almost ideals of semigroups. An almost -ideal is a generalization of -ideals and a generalization of almost one-sided ideals. We investigate properties of almost -ideals of semigroups. Moreover, we define fuzzy almost -ideals of semigroups and give relationships between almost -ideals and fuzzy almost -ideals.


Introduction and preliminaries
This notion of (m, n)-ideals of semigroups was first introduced and studied by Lajos in [1]. He investigated remarkable properties of (m, n)-ideals of semigroups in [2][3][4][5][6]. Let m and n be non-negative integers. A subsemigroup A of a semigroup S is called an (m, n)-ideal of S if A m SA n ⊆ A. Note that a left ideal of a semigroup S is a (0,1)-ideal of S and a right ideal of S is a (1,0)-ideal of S. An (m, n)-ideal is a one of generalizations of one-sided ideals. Furthermore, the theory of (m, n)-ideals in other structures have also been studied by many authors, for example, (m, n)-ideals in ordered semigroups were studied by Bussaban and Changphas in [7] and in LA-semigroups were studied by Akram et al. in [8], etc. In [9], Omidi and Davaz defined (m, n)-hyperideals and (m, n)-bi-hyperideals in ordered semihyperrings and investigate some of their related properties. Recently, Khan and Mahboob characterized (m, n)-filters of (m, n)-regular ordered semigroups in terms of its prime generalized (m, n)-ideals in [10].
In 1965, Zadeh introduced the fundamental fuzzy set concept in [11]. Since then, fuzzy sets are now applied in various fields. A fuzzy subset of S is a function from S into the closed interval [0, 1]. For any two fuzzy subsets f and g of S, (1) f ∩ g is a fuzzy subset of S defined by for all x ∈ S, (2) f ∪ g is a fuzzy subset of S defined by for all x ∈ S and For a fuzzy subset f of S, the support of f is defined by The characteristic mapping of a subset A of S is a fuzzy subset of S defined by The definition of fuzzy points was given by Pu and Liu [12]. For x ∈ S and α ∈ (0, 1], a fuzzy point x α of a set S is a fuzzy subset of a set S defined by Some interesting topics of fuzzy points were studied in [13][14][15]. Let F(S) be the set of all fuzzy subsets in a semigroup S. The semigroup S itself is a fuzzy subset of S such that S(x) = 1 for all x ∈ S, denoted also by S. For each f , g ∈ F(S), the product of f and g is a fuzzy subset f • g defined as follows: for all x ∈ S. Then F(S) is a semigroup with the prod-uct°. An introductory definition of left, right, two-sided almost ideals of semigroups were launched in 1980 by Grosek and Satko [16]. They characterized these ideals when a semigroup S contains no proper left, right, twosided almost ideals in [16], and afterwards they discovered minimal almost ideals and smallest almost ideals of semigroups in [17,18], respectively. A nonempty subset A of a semigroup S is called a left almost ideal of S if sA ∩ A = ∅ for any s ∈ S. A right almost ideal of a semigroup S is defined analogously. A nonempty subset A of a semigroup S is called an almost ideal of S if sA ∩ A = ∅ and At ∩ A = ∅ for all s, t ∈ S. In 1981, Bogdanovic [19] introduced the notion of almost bi-ideals in semigroups by using the concepts of almost ideals and bi-ideals of semigroups. Likewise, Wattanatripop, Chinram and Changphas examined quasi-almost-ideals of semigroups and gave properties of quasi-almost-ideals in [20]. Furthermore, they defined fuzzy almost ideals of semigroups in [20] and fuzzy almost bi-ideals of semigroups in [21] and provided relationship between almost ideals and fuzzy almost ideals of semigroups.
Recently, Gaketem generalized results in [21] to study interval-valued fuzzy almost bi-ideals of semigroups in [22]. In [23], Solano, Suebsung and Chinram extended this idea to study almost ideals of n-ary semigroups. Our purpose of this paper is to define the notion of almost (m, n)-ideals of semigroups by using the concepts of (m, n)-ideals and almost ideals of semigroups and study them. Moreover, we define the notion of fuzzy almost (m, n)-ideals of semigroups and give relationships between almost (m, n)-ideals and fuzzy almost (m, n)-ideals of semigroups.

Almost (m, n)-ideals
Let m and n be non-negative integers. Let A be a nonempty subset of a semigroup S and s ∈ S. Note that A 0 sA 0 := {s}. For k ∈ N, let A k sA 0 := A k s and A 0 sA k := sA k . Firstly, we define an almost (m, n)-ideal of semigroup by using the concepts of (m, n)-ideals defined in [1] and almost ideals of semigroups defined in [16].
Remark 2.1: The following statements hold.
(1) An almost (1, 0)-ideal of a semigroup S is a right almost ideal of S defined in [15]. (2) An almost (0, 1)-ideal of a semigroup S is a left almost ideal of S defined in [15].
is not a subsemigroup of Z 6 . Therefore, an almost (m, n)-ideal of a semigroup S need not be a subsemigroup of S and need not be an (m, n)-ideal of S.

Corollary 2.3: The union of two almost (m, n)-ideals of a semigroup S is an almost (m, n)-ideal of S.
Proof: Let A 1 and A 2 be any two almost (m, n)-ideals of S. Then B ⊆ S {a } for some a ∈ S. By Theorem 2.2, S {a } is also an almost (m, n)-ideal of S, this is contradiction. Therefore S has no proper almost (m, n)-ideal. Theorem 2.6: Let S be a semigroup such that |S| > 1 and a ∈ S. If S {a} is not an almost (m, n)-ideal of S, then at least one of them is true.

Fuzzy almost (m, n)-ideals
In this section, we define and study fuzzy almost (m, n)ideal and give relationships between fuzzy almost (m, n)-ideals and almost (m, n)-ideals. Let m and n be non-negative integers. Let f be a fuzzy subset and (x) α be a fuzzy point of a semigroup S. Note that f 0 := S and

Proposition 3.1: Let f,g and h be fuzzy subsets of S.
(1) If f ⊆ g, then f n ⊆ g n for all n ∈ N ∪ {0}. ( Proof: The proof is straightforward.

Definition 3.2:
A fuzzy subset f of a semigroup S is called a fuzzy almost (m, n) This implies that f is a fuzzy almost (m, n)

Proposition 3.3: Let f be a fuzzy almost (m, n)-ideal of S and g be a fuzzy subset of S such that f ⊆ g. Then g is a fuzzy almost (m, n)-ideal of S.
Proof: Assume that f is a fuzzy almost (m, n)-ideal of S and g is a fuzzy subset of S such that f ⊆ g. Let (x) α be a fuzzy point in S. We have Therefore, g is a fuzzy almost (m, n)-ideal of S.

Corollary 3.4: Let f and g be fuzzy almost (m, n)-ideals of S. Then f ∪ g is a fuzzy almost (m, n)-ideal of S.
Proof: Since f ⊆ f ∪ g, by Proposition 3.3, f ∪ g is a fuzzy almost (m, n)-ideal of S.
Note that in the proof of Corollary 3.4 is true if f or g is a fuzzy almost (m, n)-ideal of S.
We have f and g are fuzzy almost (1,0)-ideals of Z 6 but f ∩ g is not a fuzzy almost (1,0)-ideal of Z 6 . Example 3.5 implies that, in general, the intersection of two fuzzy almost (m, n)-ideals of S need not be a fuzzy almost (m, n)-ideal of S.
Note that for a subset A of S, define A 0 := S.

Lemma 3.6: Let A be a subset of S and n ∈ N ∪ {0}. Then (C A ) n = C A n .
Proof: The proof is straightforward.

Theorem 3.7: Let A be a nonempty subset of a semigroup S. Then A is an almost (m, n)-ideal of S if and only if C A is a fuzzy almost (m, n)-ideal of S.
Proof: Assume that A is an almost (m, n)-ideal of S. Then A m sA n ∩ A = ∅ for all s ∈ S. Let s ∈ S and α ∈ (0, 1]. Thus there exists x ∈ A m sA n ∩ A. So By Lemma 3.6, we have Conversely, assume that C A is a fuzzy almost (m, n)ideal of S. Let s ∈ S and α ∈ (0, 1]. Thus Hence, x ∈ A m sA n ∩ A. Eventually, A m sA n ∩ A = ∅.

Theorem 3.8: Let f be a fuzzy subset of S. Then f is a fuzzy almost (m, n)-ideal of S if and only if supp(f ) is an almost (m, n)-ideal of S.
Proof: Assume that f is a fuzzy almost (m, n)-ideal of S. Let x ∈ S. Then for any α ∈ (0, 1], we have Thus, there exists y ∈ S such that So, f (y) = 0 and y = a 1 a 2 · · · a m xb 1 b 2 · · · b n for some a 1 , a 2 , . . . , a m , b 1 , b 2 This implies that a 1 , a 2 , . . . , a m , b 1 , b 2 , . . . , b n , y ∈ supp(f ). Thus, and C supp(f ) (y) = 0. Hence, a fuzzy almost (m, n) Conversely, assume that supp(f ) is an almost (m, n)ideal of S. By Theorem 3.7, C supp(f ) is a fuzzy almost (m, n)-ideal of S. Let (x) α be a fuzzy point in S. Then Then there exists y ∈ S such that Hence, Then there exist a 1 , a 2 , . . . , a m , b 1 ,  b 2 , . . . , b n ∈ supp(f ) and y = a 1 a 2 · · · a m xb 1 b 2 · · · b n . Thus Therefore, Consequently, f is a fuzzy almost (m, n)-ideal of S.

Minimal almost (m, n)-ideals and minimal fuzzy almost (m, n)-ideals
In this subsection, we give relationship between minimal almost (m, n)-ideals and minimal fuzzy almost (m, n)-ideals.

Definition 3.9:
A fuzzy almost (m, n)-ideal f is called minimal if for all nonzero fuzzy almost (m, n)-ideals g of S such that g ⊆ f , we have supp(f ) = supp(g).

Theorem 3.10: Let S be a non-empty subset of a semigroup S. Then A is a minimal almost (m, n)-ideal of S if and only if C A is a minimal fuzzy almost (m, n)-ideal of S.
Proof: Assume that A is a minimal almost (m, n)-ideal of S. By Theorem 3.7, C A is a fuzzy almost (m, n)-ideal of S. Let f be a fuzzy almost (m, n)-ideal of S such that Conversely, assume that C A is a minimal fuzzy almost (m, n)-ideal of S. Let B be an almost (m, n)-ideal of S such that B ⊆ A. Then C B is a fuzzy almost (m, n)-ideal of S such that C B ⊆ C A . Hence, B = supp(C B ) = supp(C A ) = A. Therefore, A is minimal.  (m, n)-ideals and prime fuzzy  almost (m, n)

-ideals
In this subsection, we give relationship between prime almost (m, n)-ideals and prime fuzzy almost (m, n)ideals. Definition 3.12: Let S be a semigroup.
(1) An almost (m, n)-ideal A of S is called prime if for all x, y ∈ S, xy ∈ A implies x ∈ A or y ∈ A.

Theorem 3.13: Let A be a non-empty subset of S. Then A is a prime almost (m, n)-ideal of S if and only if C A is a prime fuzzy almost (m, n)-ideal of S.
Proof: Assume that A is a prime almost (m, n)-ideal of S. By Theorem 3.7, C A is a fuzzy almost (m, n)-ideal of S. Let x, y ∈ S. We consider two cases: Thus, C A is a prime fuzzy almost (m, n)-ideal of S. Conversely, assume that C A is a prime fuzzy almost (m, n)-ideal of S. By Theorem 3.7, A is an almost (m, n)ideal of S. Let x, y ∈ S such that xy ∈ A. Then C A (xy) = 1.
By assumption, Thus, A is a prime almost (m, n)-ideal of S.

Semiprime almost (m, n)-ideals and semiprime fuzzy almost (m, n)-ideals
In this subsection, we give relationship between semiprime almost (m, n)-ideals and semiprime fuzzy almost (m, n)-ideals. Definition 3.14: Let S be a semigroup.
(1) An almost (m, n)-ideal A of S is called semiprime if for all x ∈ S, x 2 ∈ A implies x ∈ A. (2) A fuzzy almost (m, n)-ideal f is called semiprime if for all x ∈ S, f (x 2 ) ≤ f (x).

Theorem 3.15: Let A be a non-empty subset of S. Then A is a semiprime almost (m, n)-ideal of S if and only if C A is a semiprime fuzzy almost (m, n)-ideal of S.
Proof: Assume that A is a semiprime almost (m, n)-ideal of S. By Theorem 3.7, C A is a fuzzy almost (m, n)-ideal of S. Let x ∈ S. We consider two cases: Case 1: x 2 ∈ A. Then x ∈ A. So, C A (x) = 1. Hence, C A (x) ≥ C A (x 2 ).
Case 2: x 2 ∈ A. Then C A (x 2 ) = 0 ≤ C A (x). Thus, C A is a semiprime fuzzy almost (m, n)-ideal of S. Conversely, assume that C A is a semiprime fuzzy almost (m, n)-ideal of S. By Theorem 3.7, A is an almost (m, n)-ideal of S. Let x ∈ S such that x 2 ∈ A. Then C A (x 2 ) = 1. By assumption, C A (x 2 ) ≤ C A (x). Since C A (x 2 ) = 1, C A (x) = 1. Hence, x ∈ A. Thus, A is a semiprime almost (m, n)-ideal of S.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This paper was supported by Algebra and Applications Research Unit, Prince of Songkla University.