EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6259 ISSN 1307-5543 – ejpam.com Published by New York Business Global Almost n-ary Subsemigroups and Fuzzy Almost n-ary Subsemigroups of n-ary Semigroups Ronnason Chinram1, Pattarawan Singavananda2,∗ 1 Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla 90110, Thailand 2 Mathematics Program, Faculty of Science and Technology, Songkhla Rajabhat University, Songkhla 90000, Thailand Abstract. An n-ary semigroup is a non-empty set with an associative n-ary operation. Semi- groups and ternary semigroups are special cases of n-ary semigroups where n = 2 and n = 3, respectively. In this study, we introduce and explore the notions of almost n-ary subsemigroups and their fuzzy counterparts, termed fuzzy almost n-ary subsemigroups, within the framework of n-ary semigroups. Moreover, we demonstrate certain relationships between almost n-ary subsemi- groups and fuzzy almost n-ary subsemigroups. 2020 Mathematics Subject Classifications: 20N15, 03E72 Key Words and Phrases: n-ary semigroups, almost n-ary subsemigroups, fuzzy almost n-ary subsemigroups, minimal, prime, semiprime. 1. Introduction A fuzzy subset, also known as a fuzzy set, is a generalization of the classical set. A fuzzy set is represented by a membership function of all elements in a universal set assigning values in the closed interval [0, 1]. The concept of fuzzy sets was first introduced by Zadeh [1] in 1965. Zadeh’s pioneering ideas have found widespread applications across various fields, including mathematics, computer science, and engineering. The concept of fuzzy sets has been extensively applied in the study of various algebraic structures. The generalization of binary algebraic structures to n-ary structures was first initiated by Kasner [2] in 1904. In this paper, we focus on n-ary semigroups. Notably, semigroups and ternary semigroups arise as special cases of n-ary semigroups for n = 2 and n = 3, respectively. The notion of n-ary semigroups has its origins in the investigation of algebraic structures that extend the classical frameworks of semigroups and ternary semigroups. However, an n-ary semigroup does not necessarily reduce to a semigroup or a ternary ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6259 Email addresses: ronnason.c@psu.ac.th (R. Chinram), pattarawan.pe@skru.ac.th (P. Singavananda) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) R. Chinram, P. Singavananda / Eur. J. Pure Appl. Math, 18 (3) (2025), 6259 2 of 10 semigroup. For n ≥ 3, Dudek [3] also investigated the properties of ideals and some elements of n-ary semigroups containing an idempotent. In 2018, ideals of fuzzy points n- ary semigroups were studied by Solano et al. [4]. In the next year, Couceiro and Devillet [5] showed that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and provided necessary and sufficient conditions for such a reduction to be unique. In 2020, Somsup and Leerawat [6] further investigated congruences and homomorphisms on n-ary semigroups. Later, Pornsurat and Pibaljommee [7] introduced the notions of a right regularity, a right weak regularity and a complete regularity of n-ary semigroups and characterized these regularities. In the same year, some reducibility of n-ary semigroups were investigated in [8]. In 2023, Daengsaen and Leeratanavalee [9] demonstrated that every n-ary semigroup that is both regular and intra-regular can be decomposed into a semilattice of i-simple and regular n-ary semigroups, and the reverse assertion also holds. Almost ideals on semigroups were first studied by Grosek and Satko [10] in 1980. Later, Wattanatripop et al. [11] introduced almost fuzzy ideals of semigroups and investigated relationships between almost ideals and almost fuzzy ideals. Additionally, Khamrot and Gaketem [12] and [13] introduced the concepts of bipolar fuzzy almost ideals and picture fuzzy almost ideals in semigroups, respectively. Subsequently, the concepts of almost and fuzzy almost ideals were applied to several subalgebras within various algebraic structures. For instance, almost subsemigroups and fuzzy almost subsemigroups in semigroups [14]; almost subsemigroups and fuzzy almost ternary subsemigroups in ternary semigroups [15]; almost subsemirings and fuzzy almost subsemirings in semirings [16]; and almost ternary subsemirings and fuzzy almost ternary subsemirings in ternary semirings [17] etc. This paper aims to generalize the findings presented in [14] and [15]. Basic notations and definitions are provided in Section 2. In section 3, we extend the main results. We introduce the concepts of almost n-ary subsemigroups and fuzzy almost n-ary subsemi- groups of n-ary semigroups, and present their properties. Moreover, we establish some relationships between almost n-ary subsemigroups and fuzzy almost n-ary subsemigroups. 2. Preliminaries The aim of this section is to review some notations and definitions of n-ary semigroups and fuzzy sets. 2.1. n-ary semigroups To ensure completeness, we state some definitions in the same fashion as found in [3] and [18] which are used throughout this paper. First, we recall the definition of an n-ary semigroup, where n is a positive integer such that n ≥ 2. A nonempty set A together with an n-ary operation given by f : An → A, where n ≥ 2, is called an n-ary groupoid and is denoted by the notation (A, f). According to the general convention used in the symbols of n-ary groupoids, the sequence of elements ai, ai+1, . . . , aj is denoted by aji . In the case j < i, it is the empty symbol. If ai+1 = ai+2 = · · · = ai+t = a, then we will write at instead of ai+t i+1. In this convention, we have R. Chinram, P. Singavananda / Eur. J. Pure Appl. Math, 18 (3) (2025), 6259 3 of 10 f(a1, a2, . . . , an) = f(an1 ) and f(a1, . . . , ai, a . . . , a︸ ︷︷ ︸ t , ai+t+1, . . . , an) = f(ai1, a t, ani+t+1). An n-ary groupoid (A, f) is called (i, j)-associative if f(ai−1 1 , f(an+i−1 i ), a2n−1 n+i ) = f(aj−1 1 , f(an+j−1 j ), a2n−1 n+j ) holds for all a1, a2, . . . , a2n−1 ∈ A. The n-ary operation f is called associative if the above identity holds for every 1 ≤ i ≤ j ≤ n. In the case of the n-ary operation f is associate, (A, f) is called an n-ary semigroup. A nonempty subset S of an n-ary semigroup (A, f) is called an n-ary subsemigroup of A if f(an1 ) ∈ S for all a1, a2, . . . , an ∈ S. For nonempty subsets S1, S2, . . . , Sn of A, let f(Sn 1 ) := {f(an1 ) | ai ∈ Si for all i ∈ {1, 2, . . . , n}}. If S1 = {a1}, then we write f({a1}, Sn 2 ) as f(a1, S n 2 ), and similarly in another case such as we write f({a1}, Sn−1 2 , {an}) as f(a1, Sn−1 2 , an) and so on. For any subset S of A, we let f(Sn) = {f(an1 ) | a1, a2, . . . , an ∈ S} and we let f(An) = {f(an1 ) | a1, a2, . . . , an ∈ A}. 2.2. Fuzzy subsets A fuzzy subset of a set A is defined as a membership function from A into the closed unit interval [0, 1]. We now recall some notations in fuzzy sets, as presented in [19]. Let g and h be two fuzzy subsets of a nonempty set A. 1. The intersection of g and h, denoted by g ∩ h, is a fuzzy subset of A defined by (g ∩ h)(a) = min{g(a), h(a)} for all a ∈ A. 2. The union of g and h, denoted by g∪h, is a fuzzy subset of A defined by (g∪h)(a) = max{g(a), h(a)} for all a ∈ A. 3. g ⊆ h if g(a) ≤ h(a) for all a ∈ A. For a fuzzy subset g of A, the support of g is defined by supp(g) = {a ∈ A | g(a) ̸= 0}. The characteristic mapping of a subset S of A is a fuzzy subset of A defined by χS(a) = { 1 a ∈ S, 0 a /∈ S. R. Chinram, P. Singavananda / Eur. J. Pure Appl. Math, 18 (3) (2025), 6259 4 of 10 A fuzzy subset g of an n-ary semigroup A is called a fuzzy n-ary subsemigroup of A if g(an1 ) ≥ min{g(a1), g(a2), . . . , g(an)} for all a1, a2, . . . , an ∈ A. Let F(A) be the set of all fuzzy subsets in an n-ary semigroup A. Define n-ary operator f on F(A) by f(gn1 )(a) := f(g1, g2, . . . , gn)(a) =  sup a=f(an1 ) min{g1(a1), g2(a2), . . . , gn(an)} if a ∈ f(An), 0 otherwise, for all g1, g2, . . . , gn ∈ F(A) and a ∈ A. Proposition 1. A fuzzy subset g of an n-ary semigroup A is a fuzzy n-ary subsemigroup of A if and only if f(gn) ⊆ g. 3. Main Results 3.1. Almost n-ary subsemigroups We begin by introducing the definition of almost n-ary subsemigroups of n-ary semi- groups. Definition 1. A nonempty subset S of an n-ary semigroup A is called an almost n-ary subsemigroup of A if f(Sn) ∩ S ̸= ∅. Every n-ary subsemigroup of an n-ary semigroup A is clearly an almost n-ary sub- semigroup of A. Example 1. We consider an n-ary semigroup N under the usual n-ary multiplication of integers. Let S = {2, 2n} and T = {2n, 2n2}. Clearly, S and T are almost n-ary subsemigroups but are not n-ary subsemigroups of N. However, S ∩ T = {2n} is not an almost n-ary subsemigroup of N. From Example 1, we can draw the following conclusions. (1) In general, an almost n-ary subsemigroup of an n-ary semigroup A need not be a n-ary subsemigroup of A. (2) The intersection of almost n-ary subsemigroups of an n-ary semigroup A need not be an almost n-ary subsemigroup of A. Theorem 1. Let S be an almost n-ary subsemigroup of an n-ary semigroup A. If T be a nonempty subset of A such that S ⊆ T , then T is also an almost n-ary subsemigroup of A. Proof. Let S be an almost n-ary subsemigroup of A and T be a nonempty subset of A such that S ⊆ T . Thus f(Sn)∩S ̸= ∅. Since S ⊆ T , f(Sn)∩S ⊆ f(Tn)∩T . This implies that f(Tn) ∩ T ̸= ∅. It conclude that T is an almost n-ary subsemigroup of A. The following corollary directly follows from Theorem 1. R. Chinram, P. Singavananda / Eur. J. Pure Appl. Math, 18 (3) (2025), 6259 5 of 10 Corollary 1. The union of almost n-ary subsemigroups of an n-ary semigroup A is also an almost n-ary subsemigroup of A. An element a of an n-ary semigroup A is called a selfpotent if a = f(an). Proposition 2. Let a be any element of a n-ary semigroup A. (1) If a is a selfpotent, then {a} is an almost n-ary subsemigroup of A. (2) If a is not a selfpotent, then {a, f(an)} is an almost n-ary subsemigroup of A. 3.2. Fuzzy almost n-ary subsemigroups In this subsection, we define fuzzy almost n-ary subsemigroups of n-ary semigroups and present their notable properties. A fuzzy subset g of an n-ary semigroup A is called a zero fuzzy subset if g(a) = 0 for all a ∈ A. If there exists a ∈ A such that g(a) ̸= 0, then g is called a nonzero fuzzy subset of A and we use the notation g ̸= 0. Definition 2. A fuzzy subset g of an n-ary semigroup A is called a fuzzy almost n-ary subsemigroup of A if f(gn) ∩ g is not a zero fuzzy subset of A. It is evident that a zero fuzzy subset of an n-ary semigroup A is a fuzzy n-ary sub- semigroup but not an almost fuzzy n-ary subsemigroup of A. Next, let g be a nonzero fuzzy n-ary subsemigroup of an n-ary semigroup A. By Proposition 1, we have f(gn) ⊆ g. This implies that f(gn) ∩ g = f(gn). Since g ̸= 0, there exists a ∈ A such that g(a) ̸= 0. Then f(gn)(f(an)) ̸= 0. Hence f(gn) ∩ g = f(gn) ̸= 0. Therefore, g is a fuzzy almost n-ary subsemigroup of A. We can conclude that every nonzero fuzzy n-ary subsemigroup of an n-ary semigroup A is a fuzzy almost n-ary subsemigroup of A. Example 2. We consider an n-ary semigroup N under the usual n-ary multiplication of integers. Let g and h be fuzzy subsets of N as follows: g(a) = { 0.5 if a = 2 or 2n, 0 otherwise, and h(a) = { 0.4 if a = 2n or 2n 2 , 0 otherwise. Clearly, g and h are fuzzy almost n-ary subsemigroups but are not fuzzy n-ary subsemi- groups of N. However, g ∩ h is not a fuzzy almost n-ary subsemigroup of N. From Example 2, we obtain the following conclusions. (1) A fuzzy almost n-ary subsemigroup of an n-ary semigroup A need not be a fuzzy n-ary subsemigroup of A. R. Chinram, P. Singavananda / Eur. J. Pure Appl. Math, 18 (3) (2025), 6259 6 of 10 (2) The intersection of fuzzy almost n-ary subsemigroups of an n-ary semigroup A need not be a fuzzy almost n-ary subsemigroup of A. Theorem 2. Let g be a fuzzy almost n-ary subsemigroup of an n-ary semigroup A. If h is a fuzzy subset of A such that g ⊆ h, then h is also a fuzzy almost n-ary subsemigroup of A. Proof. Since g is a fuzzy almost n-ary subsemigroup of A, f(gn) ∩ g ̸= 0. Since g ⊆ h, f(gn) ∩ g ⊆ f(hn) ∩ h. This implies that f(hn) ∩ h ̸= 0. So the proof is completed. As a direct consequence of Theorem 2, we obtain the following corollary. Corollary 2. The union of fuzzy almost n-ary subsemigroups of an n-ary semigroup A is also a fuzzy almost n-ary subsemigroup of A. 3.3. The relationships between almost n-ary subsemigroups and their fuzzifications This subsection is devoted to examining the relationships between almost n-ary sub- semigroups and fuzzy almost n-ary subsemigroups of n-ary semigroups. Theorem 3. Let S be a subset of an n-ary semigroup A. Then S is an almost n-ary subsemigroup of A if and only if χS is a fuzzy almost n-ary subsemigroup of A. Proof. Assume that S is an almost n-ary subsemigroup of A. Then S ̸= ∅ and f(Sn)∩S ̸= ∅. Hence, there exists an element a in S such that a ∈ f(Sn)∩S, Therefore, a = f(an1 ) for some a1, a2, . . . , an ∈ S and a ∈ S. It follows that f((χS) n)(a) = 1 and χS(a) = 1, which implies that (f((χS) n) ∩ χS)(a) = 1 ̸= 0. We can conclude that f((χS) n) ∩ χS ̸= 0. Hence, χS is a fuzzy almost n-ary subsemigroup of A. Conversely, assume that χS is a fuzzy almost n-ary subsemigroup of A. We have χS is a nonzero fuzzy subset of A and f((χS) n) ∩ χS ̸= 0. Then there exists an element a of A such that (f((χS) n) ∩ χS)(a) ̸= 0. So f((χS) n)(a) ̸= 0 and χS(a) ̸= 0. This implies that f((χS) n)(a) = 1 and χS(a) = 1. Hence, a ∈ f(Sn) and a ∈ S. Eventually, f(Sn) ∩ S ̸= ∅. This concludes that S is an almost n-ary subsemigroup of A. Theorem 4. Let g be a fuzzy subset of an n-ary semigroup A. Then g is a fuzzy almost n-ary subsemigroup of A if and only if supp(g) is an almost n-ary subsemigroup of A. Proof. Let g be a fuzzy almost n-ary subsemigroup of A. We have that f(gn) ∩ g is not a zero fuzzy subset of A. Thus, there exists a ∈ A such that (f(gn) ∩ g)(a) ̸= 0. Then g(a) ̸= 0 and f(gn)(a) ̸= 0. So a = f(an1 ) for some a1, a2, . . . , an ∈ A such that g(ai) ̸= 0 for all i ∈ {1, 2, . . . , n}. This implies that a1, a2, . . . , an ∈ supp(g). Therefore f((χsupp(g)) n)(a) ̸= 0 and χsupp(g)(a) ̸= 0. Hence, (f((χsupp(g)) n) ∩ χsupp(g))(a) ̸= 0. So, χsupp(g) is a fuzzy almost n-ary subsemigroup of A. By Theorem 3, supp(g) is an almost n- ary subsemigroup of A. Conversely, assume that supp(g) is an almost n-ary subsemigroup of A. By Theorem 3, we have χsupp(g) is a fuzzy almost n-ary subsemigroup of A. Thus R. Chinram, P. Singavananda / Eur. J. Pure Appl. Math, 18 (3) (2025), 6259 7 of 10 f((χsupp(g)) n)∩χsupp(g) ̸= 0. Then there is a ∈ A such that (f((χsupp(g)) n)∩χsupp(g))(a) ̸= 0. Hence, f((χsupp(g)) n)(a) ̸= 0 and χsupp(g)(a) ̸= 0. Then there exist a1, a2, . . . , an ∈ supp(g) and a = f(an1 ). Therefore g(ai) ̸= 0 for all i ∈ {1, 2, . . . , n}. Hence, f(gn)(a) ̸= 0. This implies that (f(gn) ∩ g)(a) ̸= 0. Consequently, g is a fuzzy almost n-ary subsemigroup of A. An almost n-ary subsemigroup S of an n-ary semigroup A is called minimal if for any almost n-ary subsemigroup T of A such that T ⊆ S, it follows that T = S. Next, we examine the minimality of fuzzy almost n-ary subsemigroups. A fuzzy almost n-ary subsemigroup g of an n-ary semigroup A is called minimal if for any fuzzy almost n-ary subsemigroup h of A contained in g, it follows that supp(g) = supp(h). Now, we provide the relationship between minimal almost n-ary subsemigroup and their fuzzifications. Theorem 5. A nonempty subset S of an n-ary semigroup A is a minimal almost n-ary subsemigroup of A if and only if χS is a minimal fuzzy almost n-ary subsemigroup of A. Proof. Let S be a minimal almost n-ary subsemigroup of an n-ary semigroup A. By Theorem 3, we have that χS is a fuzzy almost n-ary subsemigroup of A. Suppose that g is a fuzzy almost n-ary subsemigroup of A contained in χS . By Theorem 4, supp(g) is an almost n-ary subsemigroup of A. Since g ⊆ χS , supp(g) ⊆ supp(χS) = S. Because S is minimal, we conclude that supp(g) = S = supp(χS). Therefore, χS is minimal. Conversely, suppose that χS is a minimal fuzzy almost n-ary subsemigroup of A, and let T be an almost n-ary subsemigroup of A contained in S. By Theorem 3, we have that χT is a fuzzy almost n-ary subsemigroup of A and χT ⊆ χS . Thus, T = supp(χT ) = supp(χS) = S. We conclude that S is minimal. Corollary 3. An n-ary semigroup A has no proper almost n-ary subsemigroups if and only if for all fuzzy almost n-ary subsemigroup g of A, supp(g) = A. Proof. Assume that A has no proper almost n-ary subsemigroups, and let g be a fuzzy almost n-ary subsemigroup of A. By Theorem 4, we have supp(g) is an almost n-ary subsemigroup of A. By assumption, we have supp(g) = A. To prove the converse, we let S be any almost n-ary subsemigroup of A. By Theorem 3, we have that χS is a fuzzy almost n-ary subsemigroup of A. By assumption, we get S = supp(χS) = A. We can conclude that A has no proper almost n-ary subsemigroups. Let A be an n-ary semigroup. An almost n-ary subsemigroup S of A is called prime if for all a1, a2, . . . , an ∈ A, f(an1 ) ∈ S implies ai ∈ S for some i ∈ {1, 2, . . . , n}. A fuzzy al- most n-ary subsemigroup g of A is called prime if g(f(an1 )) ≤ max{g(a1), g(a2), . . . , g(an)} for all a1, g2, . . . , an ∈ A. Next, we investigate a relationship between prime almost n-ary subsemigroups and their fuzzifications. Theorem 6. A nonempty subset S of an n-ary semigroup A is a prime almost n-ary subsemigroup of A if and only if χS is a prime fuzzy almost n-ary subsemigroup of A. R. Chinram, P. Singavananda / Eur. J. Pure Appl. Math, 18 (3) (2025), 6259 8 of 10 Proof. Let S be any prime almost n-ary subsemigroup of A. By Theorem 3, we have that χS is a fuzzy almost n-ary subsemigroup of A. Let a1, a2, . . . , an be any n elements in A. If f(an1 ) ∈ S, then ai ∈ S for some i ∈ {1, 2, . . . , n} because S is prime. Then χS(ai) = 1 for some i ∈ {1, 2, . . . , n}. So χS(f(a n 1 )) ≤ 1 = max{χS(a1), χS(a2), . . . , χS(an)}. If f(an1 ) ̸∈ A, then χS(f(a n 1 ) = 0 ≤ max{χS(a1), χS(a2), . . . , χS(an)}. By both cases, we can conclude that χS(f(a n 1 ) ≤ max{χS(a1), χS(a2), . . . , χS(an)}. Therefore, χS is a prime fuzzy almost n-ary subsemigroup of A. To prove the converse, suppose that χS is a prime fuzzy almost n-ary subsemigroup of A. By Theorem 3, we have that S is an almost n-ary semigroup of A. Let a1, a2, . . . , an be any n elements in A such that f(an1 ) ∈ S. Thus, χS(f(a n 1 )) = 1. By assumption, we have that 1 = χS(f(a n 1 )) ≤ max{χS(a1), χS(a2), . . . , χS(an)}. Hence, max{χS(a1), χS(a2), . . . , χS(an)} = 1. We can conclude that χS(ai) = 1 for some i ∈ {1, 2, . . . , n}. Thus ai ∈ S for some i ∈ {1, 2, . . . , n}. Therefore, S is a prime almost n-ary subsemigroup of A. Let A be an n-ary semigroup. An almost n-ary subsemigroup S of A is said to be semiprime if for all a ∈ A, f(an) ∈ S implies a ∈ S. A fuzzy almost n-ary subsemigroup g of A is said to be semiprime if g(f(an)) ≤ g(a) for all a ∈ A. It is clear that every prime almost n-ary subsemigroup of A is semiprime. Similarly, every prime fuzzy almost n-ary subsemigroup of A is also semiprime. Finally, we present the relationship between semiprime almost n-ary subsemigroups and their fuzzifications. Theorem 7. A nonempty subset S of an n-ary semigroup A is a semiprime almost n-ary subsemigroup of A if and only if χS is a semiprime fuzzy almost n-ary subsemigroup of A. Proof. Let S be a semiprime almost n-ary subsemigroup of A. By Theorem 3, χS is a fuzzy almost n-ary subsemigroup of A. Let a ∈ A. If f(an) ∈ S, then a ∈ S because S is semiprime. Thus χS(a) = 1. Hence, χS(f(a n)) ≤ χS(a). If f(an) ̸∈ S, then χS(f(a n)) = 0 ≤ χS(a). In both cases, we conclude that χS(f(a n)) ≤ χS(a) for all a ∈ A. Therefore, χS is a semiprime fuzzy almost n-ary subsemigroup of A. Conversely, suppose that χS is a semiprime fuzzy n-ary subsemigroup of A. By Theorem 3, we have that S is an almost n-ary subsemigroup of A. Let a be an element in A such that f(an) ∈ S. Then χS(f(a n)) = 1. Since χS is semiprime, we have χS(f(a n)) ≤ χS(a). It follows that χS(a) = 1, and hence a ∈ S. Consequently, S is a semiprime almost n-ary subsemigroup of A. 4. Conclusion In this paper, we introduce the notions of almost n-ary subsemigroups and fuzzy al- most n-ary subsemigroups of n-ary semigroups. We prove that every n-ary semigroup is also an almost n-ary semigroup; however, the converse does not hold in general. Moreover, we show that the union of two almost n-ary subsemigroups is also an almost n-ary sub- semigroup. However, the same does not generally hold for their intersection. Similarly, we have that the union of two fuzzy almost n-ary subsemigroups is also a fuzzy almost n-ary R. Chinram, P. Singavananda / Eur. J. Pure Appl. 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