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Some New Directions in Soft (Fuzzy) Hypermodules

Some New Directions in Soft (Fuzzy) Hypermodules FUZZY INFORMATION AND ENGINEERING 2022, VOL. 14, NO. 2, 167–181 https://doi.org/10.1080/16168658.2022.2119052 a b M. Norouzi and R. Ameri a b Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran; School of Mathematics, Statistics and Computer Science, Collage of Sciences, University of Tehran, Tehran, Iran ABSTRACT ARTICLE HISTORY Received 3 May 2019 We introduce and study some new directions on soft hypermodules Revised 18 June 2021 and soft fuzzy hypermodules. In this regard, we apply soft set the- Accepted 23 August 2022 ory to hypermodules to introduce the classes of soft hypermodules and soft fuzzy hypermodules and obtain their basic properties. In KEYWORDS particular, we study the connection between soft hypermodules and Soft set; soft hypermodule; soft fuzzy hypermodules by associated (fuzzy) hyperoperations and soft fuzzy hyermodule obtain some related basic results. MATHEMATICS SUBJECT CLASSIFICATIONS 2010 20N20; 08A72 1. Introduction Nowadays, algebraic hyperstructures theory was first introduced by Marty in 1934 ([1]), is one of well-known fields of research in the context of algebra. Important applications to sev- eral domains, such as geometry, groups theory, graph theory, fuzzy sets, automata, coding theory, artificial intelligence, etc, prove this claim. Some reviews of this theory can be found in Refs. [2–5]. One of the applications of fuzzy sets theory ([6]) to algebra, initiated by Rosenfeld ([7]), is fuzzy hyperstructures. Also, one of the directions of research in the study of hyperstructures is their connections with fuzzy sets. One of the connections is established by the associa- tion of a fuzzy set with each pair of elements of a set which was introduced by Corsini and TofaninRef.[8]. This idea was extended to fuzzy semihypergroups in [9], fuzzy hyperrings and fuzzy hypermodules in [10,11], fuzzy transposition hypergroups and fuzzy topologi- cal hypergroupoids in Refs. [12,13], and fuzzy hyperalgebras in Ref. [14]. Also some other connections between fuzzy sets and hyperstructures can be seen in [15–19]. Molodtsov in Ref. [20] proposed soft sets theory for dealing with uncertainties in many areas such as economics, engineering, environmental sciences, medical sciences and social sciences, which cannot be deal with by classical methods because classical methods have inherent difficulties. Then Maji et al. in Ref. [21] introduced several operations on soft sets. Aktaş and Çogman ˇ in Ref. [22] defined soft groups and obtained the main properties of these groups. Feng et al. in Ref. [23] defined soft semirings and soft ideals on soft semirings. Moreover, the concepts of soft rings and soft modules defined by Acar et al. in Ref. [24] and Qiu-Mei Sun et al. in [25], respectively. Also, soft sets theory was discussed and studied CONTACT M. Norouzi m.norouzi65@yahoo.com, m.norouzi@ub.ac.ir © 2022 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 168 M. NOROUZI AND R. AMERI in connection with hypergroupoids [26], polygroups (isomorphism theorems) [27], semi- hyperrings (prime soft hyperideals, regularity criterion and relationship with m-systems) [28], and -hypermodules (isomorphism and fuzzy isomorphism theorems) [29], and also in topological spaces [30]. Now, in this paper, we introduce a new direction on soft algebras to hypermodules. We will proceed by introducing soft hypermodules and soft fuzzy hypermodules as a generalisation of soft modules as well as fuzzy soft modules. Some basic properties are stud- ied. Moreover, we investigate the connection between soft hypermodules and soft fuzzy hypermodules by associated (fuzzy) hyperoperations and give some related basic results. 2. Preliminaries 2.1. (Fuzzy) Hypermodules We give some definitions of (fuzzy) algebraic hyperstructures which we need to develop our paper: ∗ ∗ Let H a nonempty set and letP (H) (F (H)) be the set of all nonempty subsets (nonzero ∗ ∗ fuzzy subsets) of H.A (fuzzy) hyperoperation on H is a map ‘◦ : H × H −→ P (H)(F (H)), and the couple (H, ◦) is called a (fuzzy) hypergroupoid. A hypergroupoid (H, ◦) is called a (fuzzy) semihypergroup if for all x, y, z ∈ H,wehave (x ◦ y) ◦ z = x ◦ (y ◦ z), which means that u ◦ z = x ◦ v, (where for all μ ∈ F (H) and A ⊆ H,wehave u∈x◦y v∈y◦z (x ◦ μ)(r) = ((x ◦ t)(r) ∧ μ(t)) and (x ◦ A)(r) = (x ◦ a)(r) t∈H a∈A for all r ∈ H. Also, for two nonzero fuzzy subsets μ and λ of fuzzy semihypergroup (H, ◦),we have (μ ◦ λ)(t) = (μ(p) ∧ (p ◦ q)(t) ∧ λ(q)), for all t ∈ H). We say that (fuzzy) semi- p,q∈H hypergroup (H, ◦) is a (fuzzy) hypergroup if x ◦ H = H = H ◦ x (x ◦ H = χ = H ◦ x) for all x ∈ H. A commutative (fuzzy) hypergroup (H, ◦) is called canonical if (i) there exists e ∈ H,suchthat e ◦ x ={x} ((e ◦ x)(x)> 0), for every x ∈ H; (ii) for all x ∈ H there exists a unique x ∈ H,suchthat e ∈ x ◦ x ((x ◦ x )(e)> 0); (iii) x ∈ y ◦ z ((y ◦ z)(x)> 0) implies y ∈ x ◦ z ((x ◦ z )(y)> 0). Definition 2.1: The triple (R, , ◦) is a (fuzzy) hyperring,if (1) (R, ) is a commutative (fuzzy) hypergroup; (2) (R, ◦) is a (fuzzy) semihypergroup; (3) ‘°’ is distributive over ‘ ’. Definition 2.2: Let (R, , ◦) be a (fuzzy) hyperring. A nonempty set M, endowed with a (fuzzy) hyperoperation ‘+’, and external (fuzzy) hyperoperation ‘·’ is called a left (fuzzy) hypermodule over (R, , ◦) if the following conditions hold: (1) (M, +) is a commutative (fuzzy) hypergroup; ∗ ∗ (2) · : R × M −→ P (M)(F (M)) is such that for all a, b of M and r, s of R we have (i) r · (a + b) = (r · a) + (r · b); FUZZY INFORMATION AND ENGINEERING 169 (ii) (r s) · a = (r · a) + (s · a); (iii) (r ◦ s) · a = r · (s · a). A nonempty subset N of a the hypermodule M is called a subhypermodule of the hyper- module (M, +, ·),if (N, +) is a hypergroup and R · N ∈ P (N). Also, a nonempty subset N of a fuzzy hypermodule M is called a subfuzzy hypermodule if for all x, y ∈ N and r ∈ R,we have: (1) (x + y)(t)> 0 implies that t ∈ N; (2) x + N = χ ; (3) (r · x)(t)> 0 implies that t ∈ N. Example 2.1: Let R = [0, 1] and define the hyperoperation ⊕ for all x, y ∈ R by max max{x, y} , x = y x ⊕ y = max [0, x] x = y Then, (R, ⊕ , ·) is a hyperring (Krasner hyperring, see in Ref. [31]) where ‘·’isordinary max multiplication on real numbers. Also, I = [0, 0.5] is a hyperideal of R. Now, define a ◦ b = (a · b) ⊕ I for a, b ∈ R. Then, (R, ⊕ , ◦) is a hyperring. Set R/I ={r + I | r ∈ R}, where max max “+” is ordinary additive, and define the following hyperoperations on R/I as follows: (a + I)  (b + I) ={c + I | c ∈ a ⊕ b}, r  (z + I) ={t + I | t ∈ r ◦ z}. max Then, (R/I,,) is a hypermodule over the hyperring (R, ⊕ , ◦),byRef.[31]. max Example 2.2 ([11]): Let (M, +, ·) be a module over a ring (R, +, ·) without unity. Define the following fuzzy hyperoperations for all a, b ∈ M and r, s ∈ R: 1/2, if t = ra r  s = χ , r  s = χ , a ⊕ b = χ , (r a)(t) = {r+s} {rs} {a+b} 0, otherwise Then (M, ⊕, ) is a fuzzy hypermodule over the fuzzy hyperring (R,,). Let (M , + , · ) and (M , + , · ) be two (fuzzy) hypermodules over hyperring R. The map 1 1 1 2 2 2 f : M −→ M is called a (strong) homomorphism of (fuzzy) hypermodules if for all x, y ∈ 1 2 M and r ∈ R,wehave f (x + y) ⊆ (≤)f (x) + f (y) and f (r · x) ⊆ (≤)r · f (x) (f (x + y) = 1 1 2 1 2 1 f (x) + f (y) and f (r · x) = r · f (x)). 2 1 2 Also, we recall that if μ , μ are fuzzy subsets on M, then we say that μ is smaller than 1 2 1 μ and we denote μ ≤ μ if and only if for all x ∈ M,wehave μ (x) ≤ μ (x). Also, let 2 1 2 1 2 f : M −→ M be a map and μ be a fuzzy subset on M . Then we define f (μ) : M −→ [0, 1], 1 2 1 2 as follows: −1 (f (μ))(t) = μ(r),if f (t) =∅ −1 r∈f (t) otherwise we consider (f (μ))(t) = 0. 170 M. NOROUZI AND R. AMERI 2.2. A Connection Between Hypermodules and Fuzzy Hypermodules We recall a connection between fuzzy hypermodules and hypermodules, using the p-cuts of fuzzy sets for p ∈ [0, 1]. By Ref. [11], a structure (M, ⊕, ) is a fuzzy hypermodule over a fuzzy hyperring (R,,) if and only if (M, ⊕ , ) is a hypermodule over the hyperring p p (R, , ), for all p ∈ [0, 1], where p p x ⊕ y ={t ∈ M | (x ⊕ y)(t) ≥ p}, r  s ={u ∈ R | (r  s)(u) ≥ p} p p r x ={z ∈ M | (r x)(z) ≥ p}, r  s ={v ∈ R | (r  s)(v) ≥ p} p p for all x, y ∈ M and r, s ∈ R. Also, according to Refs. [10,11], with every fuzzy hypermodule (M, ⊕, ) over a fuzzy hyperring (R,,), we can associate a hypermodule structure (M, +, ·) over a hyperring (R, , ◦), where x + y ={t ∈ M | (x ⊕ y)(t)> 0}, r s ={u ∈ R | (r  s)(u)> 0} r · x ={z ∈ M | (r x)(z)> 0}, r ◦ s ={v ∈ R | (r  s)(v)> 0} for all x, y ∈ M and r, s ∈ R. On the other hand, let (M, +, ·) be a hypermodule over hyperring (R, , ◦). Consider the following fuzzy hyperoperations for a, b ∈ M and r, s ∈ R a ⊕ b = χ , r a = χ , r  s = χ and r  s = χ . a+b r·a r s r◦s Then (M, ⊕, ) is a fuzzy hypermodule over the fuzzy hyperring (R,,),byRef.[11]. We recall the following theorem regarding connection between subhypermodules and subfuzzy hypermodules and also homomorphism of hypermodules and fuzzy hypermod- ules in Ref. [11]: Theorem 2.1: If (M , ⊕ , ) and (M , ⊕ , ) are fuzzy hypermodules and (M , + , · ) and 1 1 1 2 2 2 1 1 1 (M , + , · ) are the associated hypermodules, then a map f : M −→ M is a homomorphism 2 2 2 1 2 of hypermodules, if f is a homomorphism of fuzzy hypermodules. Theorem 2.2: (1) If N is a subfuzzy hypermodule of (M, ⊕, ) over (R,,),thenNisa subhypermodule of associated hypermodule (M, +, ·) over (R, , ◦). (2) K is a subhypermodule of (M, +, ·) over (R, , ◦) if and only if N is a subhypermodule of associated fuzzy hypermodule (M, ⊕, ) over (R,,). Theorem 2.3: If M and M are hypermodules, then the map f : M −→ M is a homomor- 1 2 1 2 phism of hypermodules if and only if it is a homomorphism of associated fuzzy hypermodules. 2.3. Soft Sets Now, we briefly review some notions concerning soft sets. Let X be an initial universe set and E be a set of parameters. P(X) denotes the power set of X and A ⊆ E. Then, F is called a soft set over X, where F is a mapping given by F : A −→ P(X). In fact, a soft set over X is a parameterised family of subsets of the universe X.For e ∈ A, F(e) may be considered as the set of e-approximate elements of the soft set F . Note to the following example: A FUZZY INFORMATION AND ENGINEERING 171 Example 2.3 ([27]): Consider a soft set F , which describes the attractiveness of houses that one is considering for purchase. Suppose that there are six houses in the universe X,given by X ={h , h , h , h , h , h } and E ={e , e , e , e , e } is a set of decision param- 1 2 3 4 5 6 1 2 3 4 5 eters, where e (i = 1, 2, 3, 4, 5) stand for the parameters ‘expensive’, ‘beautiful’, ‘wooden’, ‘cheap’ and ‘in green surroundings’, respectively. Consider the mapping F by F(e ) ={h , h }, 1 2 4 F(e ) ={h , h }, F(e ) ={h , h , h }, F(e ) ={h , h , h } and F(e ) ={h }. The soft set F is 2 1 3 3 3 4 5 4 1 3 5 5 1 E a parameterised family {F(e ) | 1 ≤ i ≤ 5} of subsets of the set X, and can be viewed as a collection of approximations: F ={expensive houses ={h , h }, beautiful houses ={h , h }, wooden houses ={h , h , E 2 4 1 3 3 4 h }, cheap houses ={h , h , h }, in green surroundings houses ={h }}. 5 1 3 5 1 Example 2.4 ([26]): Consider two universes X and Y and an arbitrary relation, R,from X to Y. Define a set-valued function F : X −→ P(Y) by F(x) ={y ∈ Y | (x, y) ∈ R; x ∈ X}. Obviously, F is a soft set over Y. Definition 2.3: Let F and G be two soft sets over X. G is called a soft subset of F and A B B A denoted by G ⊆ F ,if B ⊆ A and G(x) ⊆ F(x) for each x ∈ B. B A Definition 2.4: (1) Intersection of two soft sets F and G over X such that A ∩ B =∅ is A B the soft set H , where C = A ∩ B and H(c) = F(c) ∩ G(c), for all c ∈ C. This is denoted by F ∩ G = H . A B C (2) Union of two soft sets F and G over X is the soft set H , where C = A ∪ B and for all A B C c ∈ C F(c),if c ∈ A − B H(c) = G(c),if c ∈ B − A F(c) ∪ G(c),if c ∈ A ∩ B. This is denoted by F ∪ G = H . B C (3) If F and G are two soft sets, then F ∧ G is defined as H , where C = A × B and A B A B C H(a, b) = F(a) ∩ G(b), for all (a, b) ∈ C. (4) For two soft sets F and G , F ∨ G , is defined by H , where C = A × B and H(a, b) = A B A B C F(a) ∪ G(b), for all (a, b) ∈ C. For a soft set F , the set Supp(F ) ={x ∈ A | F(x) =∅} is called the support of the soft set A A F .If Supp(F ) =∅, then a soft set F is called non-null. A A A Let F and G be two soft sets over X and X , respectively, and f : X −→ X and g : A −→ B A B be two functions. Then we say that pair (f, g) is a soft function from F to G , denoted by A B (f, g) : F −→ F ,if f (F(x)) = G(g(x)) for all x ∈ A. A B The concept of soft homomorphism on groups was first introduced in Ref. [22]. Consider a soft function (f, g) from F to G . The image of F under the soft function (f, g), A B A denoted by (f, g)(F ) = f (F) , is a soft set over X defined by A B ⎨ f (F(x)),if y ∈ Img f (F)(y) = g(x)=y ∅, otherwise −1 for all y ∈ B. The pre-image of G under the soft function (f, g), denoted by (f, g) (G ) = B B −1 −1 −1 f (G) , is a soft set over X defined by f (G)(x) = f (G(g(x))) for all x ∈ A. It is clear that −1 (f, g)(F ) is a soft subset of G ,and F is a soft subset of (f, g) (G ). A B A B 172 M. NOROUZI AND R. AMERI 3. Soft Hypermodules In this section, we introduce soft hypermodules and investigate their basic properties. Also, soft homomorphisms of hypermodules are discussed, and some illustrative examples are given. Suppose that (M, +, ·) is a hypermodule over a hyperring (R, , ◦) and all soft sets are considered over the hypermodule M. Definition 3.1: Let F be a non-null soft set over a hypermodule M. Then F is called a soft A A hypermodule over M if F(x) is a subhypermodule of M, for all x ∈ Supp(F ). Example 3.1: Consider A = M ={0, 1, 2, 3} and R ={0, 1, 2} with the following hyperoper- ations: + 0123 · 0123 0 {0}{0, 1}{0, 2}{0, 3} 0 {0}{0}{0}{0} 1 {0, 1}{1}{1, 2}{1, 3} 1 {0}{1}{2}{3} 2 {0, 2}{1, 2}{2}{2, 3} 2 {0}{1}{2}{3} 3 {0, 3}{1, 3}{2, 3}{3} 012 ◦ 012 0 {0}{0, 1}{0, 2} 0 {0}{0}{0} 1 {0, 1}{1}{1, 2} 1 {0}{1}{2} 2 {0, 2}{1, 2}{2} 2 {0}{1}{2} Then (M, +, ·) is a hypermodule over hyperring (R, , ◦). Define the set-valued function F : A −→ P(M) by F(x) ={a ∈ M|∃ r ∈ R; a ∈ r · x}. Then F(0) ={0}, F(1) ={0, 1}, F(2) = {0, 2} and F(3) ={0, 3} that of all these are subhypermodules of M. Hence, F is a soft hypermodule over M. Example 3.2: Let (M, +, ·) be the Z-module Z . Suppose that N ={0, 2} which is a sub- module of M.Put, x ⊕ y ={x + y}, r x = rx + N, r s ={r + s} and r ◦ s ={rs} for all x, y ∈ M and r, s ∈ Z. Then, (M, ⊕, ) is a Z-hypermodule constructed by N (see [32,Exam- ple 2.3]). Now, set M = M ∪{a, b, c} such that {a, b, c}∩ M =∅ and consider the following hyperoperations: + 0123 ab c 0 {0}{1}{2}{3}{a}{b}{c} nx +{0, 2}, x ∈ M 1 {1}{2}{3}{0}{a}{b}{c} 2 {2}{3}{0}{1}{a}{b}{c} {a}, x = a and n · x = 3 {3}{0}{1}{2}{a}{b}{c} ⎪ {b}, x = b a {a}{a}{a}{a} M {c}{b} {c}, x = c b {b}{b}{b}{b}{c} M {a} c {c}{c}{c}{c}{b}{a} M for all n ∈ Z. Then (M , + , · ) is a Z-hypermodule by [33, Page 46]. It can be seen that M = M ∪{a}, M = M ∪{b}, M = M ∪{c}, M = M and M = N are the only proper sub- 1 2 3 4 5 hypermodules of M . Now, put I ={0, 1, 2, 3} and similar to Example 3.1, define F(x) = FUZZY INFORMATION AND ENGINEERING 173 {t ∈ M |∃ n ∈ Z; t ∈ n · x} from I to P(M ). Then, F(0) = F(2) = N and F(1) = F(3) = M which are subhypermodules of M . Hence, F is a soft hypermodule over M . Note that if F : J −→ P(M ) for J ={a, b, c}, then F is not a soft hypermodule over M , since F(a) ={a}, F(b) ={b} and F(c) ={c} which are not subhypermodule in M . Notice that the intersection of two subhypermodules of a hypermodule, according to Definition 2.2, is not a subhypermodule, in general. But, if (M, +) is a canonical hyper- group in a hypermodule (M, +, ·), then the intersection of two subhypermodules of M is a subhypermodule. Hence, we have: Theorem 3.1: Let F and G be two soft hypermodules over (M, +, ·) where (M, +) is a canon- A B ical hypergroup, then (1) F ∩ G is a soft hypermodule over M, if it is non-null and A ∩ B =∅. A B (2) If A ∩ B =∅, then F ∪ G is a soft hypermodule over M. A B Proof: (1) Let H = F ∩ G , where C = A ∩ B and H(x) = F(x) ∩ G(x), for all x ∈ C.By C A B hypothesis, Supp(H ) =∅. Since the intersection of subhypermodules of M is a subhy- permodule, H(x) is a subhypermodule of M, for all x ∈ Supp(H ). Hence H is a soft C C hypermodule over M. (2) Since A ∩ B =∅, then c ∈ A ∪ B = C implies that either c ∈ A − B or c ∈ B − A for all c ∈ C.Set H = F ∪ G . Clearly, Supp(H ) =∅. By Definition 2.4, if c ∈ A − B, then H(c) = C A B C F(c) is a subhypermodule of M,and if c ∈ B − A, then H(c) = G(c) is a subhypermodule of M.Thus F ∪ G is a soft subhypermodule over M. A B Corollary 3.1: Let {(F ) |i ∈ I} be a nonempty family of soft hypermodules over (M, +, ·) where (M, +) is a canonical hypergroup, then (1) (F ) is a soft hypermodule over M, if it is non-null and A =∅. i A i i∈I i i∈I (2) If {A |i ∈ I} are pairwise disjoin, then (F ) is a soft hypermodule over M. i i A i∈I i Proof: By using Theorems 3.1, the proof is complete. Theorem 3.2: Suppose that F and G are two soft hypermodules over (M, +, ·) where (M, +) A B is a canonical hypergroup, then (1) F ∧ G is a soft hypermodule over M. A B (2) If F(x) ⊆ G(y) or G(y) ⊆ F(x) for all x ∈ Aand y ∈ B, then F ∨ G is a soft hypermodule A B over M. Proof: (1) Using Definition 2.4, we know that F ∧ G = H , where H(x, y) = F(x) ∩ G(y) A B A×B for all (x, y) ∈ A × B. Obviously, Supp(H ) = Supp(F ) × Supp(G ) =∅. Since the inter- A×B A B section of two subhypermodules is a subhypermodules, it follows that H(x, y) = F(x) ∩ G(y) is a subhypermodule of M for all (x, y) ∈ Supp(H ). Consequently, H is a soft A×B A×B hypermodules over M. 174 M. NOROUZI AND R. AMERI (2) Similar to the proof of (3), by hypothesis, for all (x, y) ∈ Supp(H ) we have H(x, y) = A×B F(x) or H(x, y) = G(y) which are subhypermodules of M. Therefore, H = F ∨ G is a soft A×B A B hypermodule of M. Corollary 3.2: For a nonempty family {(F ) |i ∈ I} of soft hypermodules over (M, +, ·) where (M, +) is a canonical hypergroup, we have (1) (F , A ) is a soft hypermodule over M. i i i∈I (2) If F (a ) ⊆ F (a ) or F (a ) ⊆ F (a ) for all i, j ∈ Iand a ∈ A , then (F , A ) is a soft hyper- i i j j j j i i i i i i i∈I module over M. Proof: By using Theorems 3.2, the proof is complete. Definition 3.2: Let F and G be soft hypermodules over two hypermodules M and M , A B respectively. Let (f, g) be a soft function from F to G .The pair (f, g) is called a homomor- A B phism of soft hypermodules, if f is a homomorphism of hypermodules. Example 3.3: Consider the hypermodules M ={0, 1, 2, 3} and M = R ={0, 1, 2} defined in Example 3.1 (every hyperring is a hypermodule over itself). Let A ={0, 1, 2} and B ={0, 2}. Consider the set-valued functions F : A −→ P(M) and G : B −→ P(M ), which are given by F(x) ={a ∈ M |∃ r ∈ R; a ∈ r · x} and G(x) ={y ∈ M | y ∈ x 0}.Itiseasytoseethat F and G are soft hypermodules over M and M , respectively. Define the function f : M −→ M , which f (0) = 0, f (1) = f (2) = 2and f (3) = 1, then f is a homomorphism of hypermod- ules. Moreover, consider the function g : A −→ B defined by g(0) = 0and g(1) = g(2) = 2. Since f (F(0)) = f ({0}) ={f (0)}={0}= G(0) = G(g(0)), f (F(1)) = f ({0, 1}) ={f (0), f (1)}={0, 2}= G(2) = G(g(1)), f (F(2)) = f ({0, 2}) ={f (0), f (2)}={0, 2}= G(2) = G(g(2)), then (f, g) is a homomorphism of soft hypermodules. Lemma 3.1: Let (f, g) be a soft function from F to G . Then A B (i) If g is onto, then Supp(f (F) ) = g(Supp(F )). B A −1 −1 (ii) Supp(f (G) ) ⊆ g (Supp(G )). A B Proof: (i) Let y ∈ Supp(f (F) ), then ∅ = f (F)(y) = f (F(x)). Hence, there exists x ∈ y=g(x) A such that y = g(x) and F(x) =∅. It implies that y ∈ g(Supp(F )) and so Supp(f (F) ) ⊆ A B g(Supp(F )). On the other hand, g(Supp(F )) ={g(x) | F(x) =∅}. Hence, y ∈ g(Supp(F )) A A A implies that y = g(x) such that F(x) =∅ for x ∈ A. Then f (F)(y) = f (F(x)) =∅ and g(x)=y so y ∈ Supp(f (F) ). Therefore, the proof of (i) is complete. −1 −1 −1 (ii) Suppose that x ∈ Supp(f (G) ). Then, f (G(g(x))) = f (G)(x) =∅. It implies that −1 y = g(x) such that G(y) =∅ for y ∈ B. Hence, x ∈ g (y) and y ∈ Supp(G ). Then, x ∈ −1 g (Supp(G )). B FUZZY INFORMATION AND ENGINEERING 175 Theorem 3.3: Let g : A −→ B be a bijective mapping and (f, g) be a strong homomorphism from F to G , two soft hypermodules over M and M , respectively. Then f (F) is a soft hyper- A B B module over M . Proof: Let y ∈ Supp(f (F) ). By Lemma 3.1 (i), there exists x ∈ Supp(F ) such that y = g(x), B A and F(x) =∅ is a subhypermodule of M. Also, since g is one to one, we have f (F)(y) = f (F(t)) = f (F(x)). y=g(t) Now, since f is a strong homomorphism, f (F(x)) is a subhypermodule of M . It follows that f (F) is a soft hypermodule over M . Theorem 3.4: Let F and G be two soft hypermodules over M and M , respectively, and (f, g) A B −1 a homomorphism from F to G . Then f (G) is a soft hypermodule over M. A B A −1 Proof: Let x ∈ Supp(f (G) ). Then g(x) ∈ Supp(G ) by Lemma 3.1(ii). Hence, G(g(x)) is a A B subhypermodule of M since G is a soft hypermodule. Now, since f is a homomorphism, −1 −1  −1 we get f (G)(x) = f (G(g(x))) is a subhypermodule of M . Therefore, f (G) is a soft hypermodule over M. Theorem 3.5: Let F ,G and H be soft hypermodules over M, M and M , respectively, and A B C (f, g) : F −→ G and (f , g ) : G −→ H be two homomorphisms of soft hypermodules. Then A B B C (f ◦ f, g ◦ g) : F −→ H is a soft (resp. strong) homomorphism of hypermodules. A C Proof: It is straightforward. Definition 3.3: Let F and G be two soft hypermodules over M. Then G is called a soft A B B subhypermodule of F if B ⊆ A and G(x) is a subhypermodule of F(x), for all x ∈ Supp(G ). A B Example 3.4: Consider hypermodule (M, +, ·) over hyperring (R, , ◦) defined in Exam- ple 3.1. Set B ={0, 2} and consider the set-valued function G : B −→ P(M) given by G(x) = {y ∈ M | y ∈ x + 0}. Then G(0) ={0} and G(2) ={0, 2} that are subhypermodules of M and so G is a soft hypermodule over M. Consider the soft hypermodule F defined in Exam- B A ple 3.1. Since B ⊂ A, Supp(G ) ={0, 2}, and clearly G(0) and G(2) are subhypermodules of F(0) and F(2), respectively, then G is a soft subhypermodule of F . B A Theorem 3.6: Let F be a soft hypermodule over (M, +, ·) where (M, +) is a canonical hyper- group, and {(F ) | i ∈ I} be a nonempty family of soft subhypermodules of F . i Ai A (1) (F ) is a soft subhypermodule of F if it is non-null and A =∅. i A A i i∈I i i∈I (2) If {A |i ∈ I} are pairwise disjoin, then (F ) is a soft subhypermodule over M. i i A i∈I i (3) (F ) is a soft subhypermodule over M. i A i∈I i (4) If F (a ) ⊆ F (a ) or F (a ) ⊆ F (a ) for all i, j ∈ Iand a ∈ A , then (F ) is a soft subhy- i i j j j j i i i i i A i∈I i permodule over M. Proof: The proof is similar to the proof of Theorems 3.1 and 3.2.  176 M. NOROUZI AND R. AMERI Theorem 3.7: Let f : M −→ M be a homomorphism of hypermodules. (i) Iffisstrong, andF and G are two soft hypermodules over M such that F is a soft A B A subhypermodule of G , then f (F) is a soft subhypermodule of f (G) . B A B (ii) If F and G are two soft hypermodules over M such that F is a soft subhypermodule of A B A −1  −1 G , then f (F ) is a soft subhypermodule of f (G ) . A B Proof: (i) Clearly, A ⊆ B.Let x ∈ Supp(f (F) ). Hence, f (F(x)) = f (F)(x) =∅ which implies that F(x) =∅. By hypothesis F(x) is a subhypermodule of G(x).Since f is a strong homo- morphism, f (F(x)) is a subhypermodule of f (G(x)) = f (G)(x) and so f (F) is a soft subhy- permodule of f (G) . −1 (ii) Suppose that F is a soft subhypermodule of G ,and x ∈ Supp(f (F )  ). Hence, A B −1  −1 f (F (x)) = f (F )(x) =∅. Then, F (x) =∅ which is a subhypermodule of G (x).Since f −1  −1  −1 is a homomorphism, f (F (x)) is a subhypermodule of f (G (x)) = f (G )(x). It follows −1  −1 that f (F )  is a soft subhypermodule of f (G ) . A B 4. Soft Fuzzy Hypermodules In this section we introduce the notion of a soft fuzzy hypermodule and present some prop- erties of them. In what follow, (M, ⊕, ) is a fuzzy hypermodule over a fuzzy hyperring (R,,) and all soft sets are considered over M. Definition 4.1: A non-null soft set F over a fuzzy hypermodule M is called a soft fuzzy hypermodule if F(x) is a subfuzzy hypermodule of M, for all x ∈ Supp(F ). Example 4.1: Consider the hypermodule (M, +, ·) over the hyperring (R, , ◦) defined in Example 3.1 and A ={0, 1, 2, 3}. Define a ⊕ b = χ , r a = χ , r  s = χ , r  s = χ a+b r·a r s r◦s for all a, b ∈ M and r, s ∈ R. Then (M, ⊕, ) is a fuzzy hypermodule over fuzzy hyperring (R,,),byRef.[11]. Consider the set-valued function F : A −→ P(M) defined by F(x) ={a ∈ M |∃r ∈ R, (r x)(a)> 0}. Then F(0) ={0}, F(1) ={0, 1}, F(2) ={0, 2} and F(3) ={0, 3} that are subfuzzy hypermod- ules of M. Hence, F is a soft fuzzy hypermodule over M. Lemma 4.1: A nonempty subset K is a subfuzzy hypermodule of fuzzy hypermodule (M, ⊕, ) if and only if K is a subhypermodule of (M, ⊕ , ) for all p ∈ [0, 1]. p p Proof: K is a subfuzzy hypermodule of (M, ⊕, ) iff (K, ⊕, ) is a fuzzy hypermodule iff (K, ⊕ , ) for all p ∈ [0, 1] is a hypermodule iff K is a subhypermodule of (M, ⊕ , ) for p p p p all p ∈ [0, 1]. Lemma 4.2: Let N and K be two subfuzzy hypermodules of (M, ⊕, ) where (M, ⊕) is a canonical fuzzy hypergroup. Then N ∩ K is a subfuzzy hypermodule of M. FUZZY INFORMATION AND ENGINEERING 177 Proof: Let x, y ∈ N ∩ K, t ∈ M and (x ⊕ y)(t)> 0. Since, x, y ∈ N, x, y ∈ K,and N and K are subfuzzy hypermodules, then t ∈ N and t ∈ K,and so t ∈ N ∩ K. Similarly, (r x)(t)> 0 implies that t ∈ N ∩ K for all r ∈ R. Also, for z ∈ N ∩ K and p ∈ (0, 1], by associated hyper- module (M, ⊕ , ),wehave p p z ⊕ (N ∩ K) (t) ≥ p ⇐⇒ t ∈ z ⊕ (N ∩ K) ⇐⇒ t ∈ (z ⊕ N) ∩ (z ⊕ K) p p ⇐⇒ t ∈ N ∩ K (by Lemma 4.1) ⇐⇒ χ (t) ≥ p. N∩K Now, consider the associated hypermodule (M, +, ·). Then z ⊕ (N ∩ K) (t) = 0 ⇔ t ∈ z + (N ∩ K) = (z + N) ∩ (z + K) = N ∩ K ⇔ χ (t) = 0. N∩K It follows that z ⊕ (N ∩ K) = χ . Therefore, the proof is complete. N∩K Theorem 4.1: Let F and G be two soft fuzzy hypermodules over M where (M, ⊕) is a canon- ical fuzzy hypergroup. Then: (1) F ∩ G is a soft fuzzy hypermodule over M, if it is non-null and A ∩ B =∅. A B (2) If A ∩ B =∅, then F ∪ G is a soft fuzzy hypermodule over M. A B (3) F ∧ G is a soft fuzzy hypermodule over M. A B (4) If F(x) ⊆ G(y) or G(y) ⊆ F(x) for all x ∈ Aandy ∈ B, then F ∨ G is a soft fuzzy hypermod- A B ule over M. Proof: By using Lemma 4.2, similar to the proof of Theorems 3.1 and 3.2, the proof is completed. Said results in Corollaries 3.1 and 3.2 are valid for soft fuzzy hypermodules, similarly. Definition 4.2: A soft fuzzy hypermodule G is said to be a soft subfuzzy hypermodule of soft fuzzy hypermodule F ,if B ⊆ A and G(x) is a subfuzzy hypermodule of F(x), for all x ∈ Supp(G ). Example 4.2: Set B ={0, 2} and consider the set-valued function G : B −→ P(M) given by G(x) ={y ∈ M | (x ⊕ 0)(y)> 0} over the fuzzy hypermodule (M, ⊕, ) and soft fuzzy hypermodule F defined in Example 4.1. Then G(0) ={0} and G(2) ={0, 2} which are sub- fuzzy hypermodules of M and also F(0) and F(2), respectively. Hence G is a soft subfuzzy hypermodule of F . Definition 4.3: The pair (f, g) is called a homomorphism of soft fuzzy hypermodules F and G over two fuzzy hypermodules M and M , respectively, if (f, g) is a soft function from F to B A G , then f is a homomorphism of fuzzy hypermodules. Example 4.3: Consider two fuzzy hypermodules (M, ⊕, ) and (M = R,,) as in Exam- ple 4.1. Set A ={0, 1, 2}, B ={0, 2} and let set-valued functions F : A −→ P(M) and G : 178 M. NOROUZI AND R. AMERI B −→ P(M ), which are given by F(x) ={a ∈ M |∃ r ∈ R; (r x)(a)> 0} and G(x) ={y ∈ M | (x  0)(y)> 0}. Then F and G are soft fuzzy hypermodules over M and M ,respec- A B tively. Let f and g as in Example 3.3. By connection of hypermodules and fuzzy hypermod- ules, it concludes that f is a homomorphism of fuzzy hypermodules. This implies that, (f, g) is a homomorphism of soft fuzzy hypermodules. Lemma 4.3: Let f : M −→ M be a homomorphism of fuzzy hypermodules and N and K sub- fuzzy hypermodules of M and M , respectively. (1) If fisstrong, thenf (N) is a subfuzzy hypermodule of M . −1 (2) f (K) is a subfuzzy hypermodule of M. Proof: (1) Suppose that f (x), f (y) ∈ f (N), t ∈ M and (f (x) ⊕ f (y))(t)> 0. Consider the associated hypermodule (M, +, ·). By Theorem 2.2 (1) and Theorem 2.1, it implies that t ∈ f (x) + f (y) = f (x + y) ⊆ f (N). Also, similar to the proof of Lemma 4.2, we can show that f (x) ⊕ f (N) = χ ,and (r f (x))(t)> 0 implies that t ∈ f (N) for r ∈ R. Similarly, (2) will be f (N) proved. By Lemma 4.3, it can be seen that Theorems 3.3, 3.4, 3.5 and 3.7 are valid for soft fuzzy hypermodules similarly. 5. Connections Between Soft Hypermodules and Soft Fuzzy Hypermodules In this section, the connections between soft hypermodules and soft fuzzy hypermodules and also their homomorphisms are investigated based on associated hyperoperations and fuzzy hyperoperations ([11]). The connections between fuzzy hyperoperations and associated hyperoperations have been considered by Sen, Ameri and Chowdhury in the context of semihypergroups and hypergroups ([9]), by Leoreanu-Fotea and Davvaz in hyperrings ([10]), and by Leoreanu- Fotea in hypermodules ([11]). The following theorems establish a similar result for soft hypermodules and soft fuzzy hypermodules. Theorem 5.1: Let F be a soft fuzzy hypermodule over fuzzy hypermodule (M, ⊕, ) on a fuzzy hyperring (R,,). Then F is a soft hypermodule over associated hypermodule (M, +, ·) on associated hyperring (R, , ◦). Proof: Since F is soft fuzzy hypermodule, F(x) is a subfuzzy hypermodule of (M, ⊕, ) for all x ∈ A. By Theorem 2.2 (1), F(x) is a subhypermodule of associated hypermodule M,for all x ∈ A. Therefore, F is a soft hypermodule over (M, +, ·). If we denote by SHM the class of all soft hypermodules and by SFHM the class of all soft fuzzy hypermodules, then we can consider the map  : SFHM −→ SHM such that  ((F , ⊕, )) = (F , +, ·). S A A FUZZY INFORMATION AND ENGINEERING 179 Example 5.1: Consider soft fuzzy hypermodule F over fuzzy hypermodule (M, ⊕, ) defined in Example 4.1. We have F(x) ={a ∈ M |∃r ∈ R; (r x)(a)> 0}={a ∈ M |∃r ∈ R; a ∈ r · x} for all x ∈ A. F is a soft hypermodule over associated hypermodule (M, +, ·) by Example 3.1. Lemma 5.1: Let (M, +, ·) be a hypermodule on (R, , ◦),and (M, ⊕, ) the associated fuzzy hypermodule on associated fuzzy hyperring (R,,). Then a ⊕ N = χ for every subhyper- a+N module N of M and for all a ∈ N. Proof: Let p ∈ (0, 1] and t ∈ M. Then (a ⊕ N)(t) ≥ p ⇔ (a ⊕ n)(t) ≥ p ⇔∃b ∈ N; χ (t) ≥ p ⇔ t ∈ a + N = N a+b n∈N ⇔ χ (t) ≥ p. Also, (a ⊕ N)(t) = 0iff t ∈ a + N = N iff χ (t) = 0. Therefore, for all p ∈ [0, 1] we have (a ⊕ N)(t) ≥ p ⇐⇒ χ (t) ≥ p. Hence, a ⊕ N = χ . N a+N The next theorem shows that we can obtain a soft fuzzy hypermodule by a soft hyper- module. Theorem 5.2: F is a soft hypermodule over a hypermodule (M, +, ·) on a hyperring (R, , ◦) if and only if F is a soft fuzzy hypermodule over an associated fuzzy hypermodule (M, ⊕, ) on associated fuzzy hyperring (R,,). Proof: By Theorem 2.2 (2), we have F is a soft hypermodule over M if and only if F(x) is a subhypermodule of M for all x ∈ A if and only if F(x) is a subfuzzy hypermodule of asso- ciated fuzzy hypermodule M for all x ∈ A if and only if F is a soft fuzzy hypermodule over associated fuzzy hypermodule M. Hence, there exists a map  : SHM −→ SFHM such that  ((F , +, ·)) = (F , ⊕, ). S S A A Example 5.2: Consider soft hypermodule G over hypermodule (M, +, ·) defined in Exam- ple 3.4. By associated fuzzy hyperoperations we have G(x) ={y ∈ M | y ∈ x + 0}={y ∈ M | χ (y) = 1}={y ∈ M | (x ⊕ 0)(y)> 0}. x+0 G is a soft fuzzy hypermodule over associated fuzzy hypermodule (M, ⊕, ) (Example 4.1). Now, we investigate the connection between homomorphism of soft hypermodules and homomorphism of soft fuzzy hypermodule. Theorem 5.3: Let (F , ⊕ , ) and (G , ⊕ , ) be two soft fuzzy hypermodules over A 1 1 B 2 2 fuzzy hypermodules (M , ⊕ , ) and (M , ⊕ , ), respectively, and also (F , + , · ) = 1 1 1 2 2 2 A 1 1 ((F , ⊕ , )) and (G , + , · ) =  ((G , ⊕ , )) be associated soft hypermodules. If S A 1 1 B 2 2 S B 2 2 180 M. NOROUZI AND R. AMERI (f, g) is a homomorphism of soft fuzzy hypermodules, then (f, g) is a homomorphism of soft hypermodule homomorphism, too. Proof: By Theorem 2.1, if f is a homomorphism of fuzzy hypermodules, then f is a homo- morphism of associated hypermodules, too. This completes the proof. Theorem 5.4: Let (F , + , · ) and (G , + , · ) be two soft hypermodules over hypermodules A 1 1 B 2 2 M and M and (F , ⊕ , ) =  ((F , + , · )) and (G , ⊕ , ) =  ((G , + , · )) be asso- 1 2 A 1 1 S A 1 1 B 2 2 S B 2 2 ciated soft fuzzy hypermodules. If (f, g) is a homomorphism of soft fuzzy hypermodule if and only if (f, g) is a homomorphism of soft hypermodules. Proof: By Theorem 2.3, the proof is complete. 6. Conclusion and Future Work We combined three theories soft sets, fuzzy sets and algebraic hyperstructures (here hyper- modules) to introduce soft fuzzy hypermodules, as a generalisation of soft fuzzy modules. We studied some basic properties of classes soft hypermodules and soft fuzzy hyper- modules. Furthermore, we introduced associated (fuzzy) hyperoperations and soft (fuzzy) homomorphisms to establish a connection between the class of soft hypermodules and soft fuzzy hypermodules. This paper provided the general tools to study soft set theory and soft fuzzy concept on algebraic hyperstructures ; therefore, it can be considered as a generalisa- tion of the works in this area (such as Refs. [26–29]). Also note that the presented results in this paper, can be easily extended for (m, n)-hyperstructures such as (m, n)-hypermodules or e-hyperstructures. Disclosure statement No potential conflict of interest was reported by the authors. Notes on Contributors Morteza Norouzi is an assistance professor in Department of Mathematics at University of Bojnord, Bojnord, Iran. He received BSc, MSc, and PhD degrees in Pure Mathematics (Algebra, Algebraic Hyperstructures) from University of Mazandaran, Babolsar, Iran (2004–2014). His research interests is in connection with algebraic hyperstructures: fundamental relations, (m, n)-ary hyperstructures, chemical hyperstructures and fuzzy algebraic hyperstructures. Reza Ameri is a professor in Department of Mathematics, School of Mathematics, Statistics and Com- puter Science, University of Tehran, Tehran, Iran. He received his M.S. in 1993 in Fuzzy Algebra, and his Ph.D. in 1997 in algebra(Algebraic Hyperstructures), from Shahid Bahonar University of Kerman, Kerman, Iran. He spent one year as visiting professor in Montreal University in Canada in 2004. His research interest is Algebra and Algebraic Hyperstructures: Category Theory, Universal Algebra and Fuzzy Mathematics. ORCID M. Norouzi http://orcid.org/0000-0001-9850-1126 FUZZY INFORMATION AND ENGINEERING 181 References iem [1] Marty F. Sur une generalization de la notion de groupe. In: 8 congres des Mathematiciens Scandinaves. Stockholm. 1934. p. 45–49. [2] Corsini P. Prolegomena of hypergroup theory. 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[32] Anvariyeh SM, Davvaz B. Strongly transitive geometric spaces associated to hypermodules. J Algebra. 2009;322:1340–1359. [33] Anvariyeh SM, Davvaz B. On the heart of hypermodules. Math Scand. 2010;106:39–49. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Some New Directions in Soft (Fuzzy) Hypermodules

Norouzi, M.; Ameri, R.
Fuzzy Information and Engineering , Volume 14 (2): 15 – Apr 3, 2022
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FUZZY INFORMATION AND ENGINEERING 2022, VOL. 14, NO. 2, 167–181 https://doi.org/10.1080/16168658.2022.2119052 a b M. Norouzi and R. Ameri a b Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran; School of Mathematics, Statistics and Computer Science, Collage of Sciences, University of Tehran, Tehran, Iran ABSTRACT ARTICLE HISTORY Received 3 May 2019 We introduce and study some new directions on soft hypermodules Revised 18 June 2021 and soft fuzzy hypermodules. In this regard, we apply soft set the- Accepted 23 August 2022 ory to hypermodules to introduce the classes of soft hypermodules and soft fuzzy hypermodules and obtain their basic properties. In KEYWORDS particular, we study the connection between soft hypermodules and Soft set; soft hypermodule; soft fuzzy hypermodules by associated (fuzzy) hyperoperations and soft fuzzy hyermodule obtain some related basic results. MATHEMATICS SUBJECT CLASSIFICATIONS 2010 20N20; 08A72 1. Introduction Nowadays, algebraic hyperstructures theory was first introduced by Marty in 1934 ([1]), is one of well-known fields of research in the context of algebra. Important applications to sev- eral domains, such as geometry, groups theory, graph theory, fuzzy sets, automata, coding theory, artificial intelligence, etc, prove this claim. Some reviews of this theory can be found in Refs. [2–5]. One of the applications of fuzzy sets theory ([6]) to algebra, initiated by Rosenfeld ([7]), is fuzzy hyperstructures. Also, one of the directions of research in the study of hyperstructures is their connections with fuzzy sets. One of the connections is established by the associa- tion of a fuzzy set with each pair of elements of a set which was introduced by Corsini and TofaninRef.[8]. This idea was extended to fuzzy semihypergroups in [9], fuzzy hyperrings and fuzzy hypermodules in [10,11], fuzzy transposition hypergroups and fuzzy topologi- cal hypergroupoids in Refs. [12,13], and fuzzy hyperalgebras in Ref. [14]. Also some other connections between fuzzy sets and hyperstructures can be seen in [15–19]. Molodtsov in Ref. [20] proposed soft sets theory for dealing with uncertainties in many areas such as economics, engineering, environmental sciences, medical sciences and social sciences, which cannot be deal with by classical methods because classical methods have inherent difficulties. Then Maji et al. in Ref. [21] introduced several operations on soft sets. Aktaş and Çogman ˇ in Ref. [22] defined soft groups and obtained the main properties of these groups. Feng et al. in Ref. [23] defined soft semirings and soft ideals on soft semirings. Moreover, the concepts of soft rings and soft modules defined by Acar et al. in Ref. [24] and Qiu-Mei Sun et al. in [25], respectively. Also, soft sets theory was discussed and studied CONTACT M. Norouzi m.norouzi65@yahoo.com, m.norouzi@ub.ac.ir © 2022 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 168 M. NOROUZI AND R. AMERI in connection with hypergroupoids [26], polygroups (isomorphism theorems) [27], semi- hyperrings (prime soft hyperideals, regularity criterion and relationship with m-systems) [28], and -hypermodules (isomorphism and fuzzy isomorphism theorems) [29], and also in topological spaces [30]. Now, in this paper, we introduce a new direction on soft algebras to hypermodules. We will proceed by introducing soft hypermodules and soft fuzzy hypermodules as a generalisation of soft modules as well as fuzzy soft modules. Some basic properties are stud- ied. Moreover, we investigate the connection between soft hypermodules and soft fuzzy hypermodules by associated (fuzzy) hyperoperations and give some related basic results. 2. Preliminaries 2.1. (Fuzzy) Hypermodules We give some definitions of (fuzzy) algebraic hyperstructures which we need to develop our paper: ∗ ∗ Let H a nonempty set and letP (H) (F (H)) be the set of all nonempty subsets (nonzero ∗ ∗ fuzzy subsets) of H.A (fuzzy) hyperoperation on H is a map ‘◦ : H × H −→ P (H)(F (H)), and the couple (H, ◦) is called a (fuzzy) hypergroupoid. A hypergroupoid (H, ◦) is called a (fuzzy) semihypergroup if for all x, y, z ∈ H,wehave (x ◦ y) ◦ z = x ◦ (y ◦ z), which means that u ◦ z = x ◦ v, (where for all μ ∈ F (H) and A ⊆ H,wehave u∈x◦y v∈y◦z (x ◦ μ)(r) = ((x ◦ t)(r) ∧ μ(t)) and (x ◦ A)(r) = (x ◦ a)(r) t∈H a∈A for all r ∈ H. Also, for two nonzero fuzzy subsets μ and λ of fuzzy semihypergroup (H, ◦),we have (μ ◦ λ)(t) = (μ(p) ∧ (p ◦ q)(t) ∧ λ(q)), for all t ∈ H). We say that (fuzzy) semi- p,q∈H hypergroup (H, ◦) is a (fuzzy) hypergroup if x ◦ H = H = H ◦ x (x ◦ H = χ = H ◦ x) for all x ∈ H. A commutative (fuzzy) hypergroup (H, ◦) is called canonical if (i) there exists e ∈ H,suchthat e ◦ x ={x} ((e ◦ x)(x)> 0), for every x ∈ H; (ii) for all x ∈ H there exists a unique x ∈ H,suchthat e ∈ x ◦ x ((x ◦ x )(e)> 0); (iii) x ∈ y ◦ z ((y ◦ z)(x)> 0) implies y ∈ x ◦ z ((x ◦ z )(y)> 0). Definition 2.1: The triple (R, , ◦) is a (fuzzy) hyperring,if (1) (R, ) is a commutative (fuzzy) hypergroup; (2) (R, ◦) is a (fuzzy) semihypergroup; (3) ‘°’ is distributive over ‘ ’. Definition 2.2: Let (R, , ◦) be a (fuzzy) hyperring. A nonempty set M, endowed with a (fuzzy) hyperoperation ‘+’, and external (fuzzy) hyperoperation ‘·’ is called a left (fuzzy) hypermodule over (R, , ◦) if the following conditions hold: (1) (M, +) is a commutative (fuzzy) hypergroup; ∗ ∗ (2) · : R × M −→ P (M)(F (M)) is such that for all a, b of M and r, s of R we have (i) r · (a + b) = (r · a) + (r · b); FUZZY INFORMATION AND ENGINEERING 169 (ii) (r s) · a = (r · a) + (s · a); (iii) (r ◦ s) · a = r · (s · a). A nonempty subset N of a the hypermodule M is called a subhypermodule of the hyper- module (M, +, ·),if (N, +) is a hypergroup and R · N ∈ P (N). Also, a nonempty subset N of a fuzzy hypermodule M is called a subfuzzy hypermodule if for all x, y ∈ N and r ∈ R,we have: (1) (x + y)(t)> 0 implies that t ∈ N; (2) x + N = χ ; (3) (r · x)(t)> 0 implies that t ∈ N. Example 2.1: Let R = [0, 1] and define the hyperoperation ⊕ for all x, y ∈ R by max max{x, y} , x = y x ⊕ y = max [0, x] x = y Then, (R, ⊕ , ·) is a hyperring (Krasner hyperring, see in Ref. [31]) where ‘·’isordinary max multiplication on real numbers. Also, I = [0, 0.5] is a hyperideal of R. Now, define a ◦ b = (a · b) ⊕ I for a, b ∈ R. Then, (R, ⊕ , ◦) is a hyperring. Set R/I ={r + I | r ∈ R}, where max max “+” is ordinary additive, and define the following hyperoperations on R/I as follows: (a + I)  (b + I) ={c + I | c ∈ a ⊕ b}, r  (z + I) ={t + I | t ∈ r ◦ z}. max Then, (R/I,,) is a hypermodule over the hyperring (R, ⊕ , ◦),byRef.[31]. max Example 2.2 ([11]): Let (M, +, ·) be a module over a ring (R, +, ·) without unity. Define the following fuzzy hyperoperations for all a, b ∈ M and r, s ∈ R: 1/2, if t = ra r  s = χ , r  s = χ , a ⊕ b = χ , (r a)(t) = {r+s} {rs} {a+b} 0, otherwise Then (M, ⊕, ) is a fuzzy hypermodule over the fuzzy hyperring (R,,). Let (M , + , · ) and (M , + , · ) be two (fuzzy) hypermodules over hyperring R. The map 1 1 1 2 2 2 f : M −→ M is called a (strong) homomorphism of (fuzzy) hypermodules if for all x, y ∈ 1 2 M and r ∈ R,wehave f (x + y) ⊆ (≤)f (x) + f (y) and f (r · x) ⊆ (≤)r · f (x) (f (x + y) = 1 1 2 1 2 1 f (x) + f (y) and f (r · x) = r · f (x)). 2 1 2 Also, we recall that if μ , μ are fuzzy subsets on M, then we say that μ is smaller than 1 2 1 μ and we denote μ ≤ μ if and only if for all x ∈ M,wehave μ (x) ≤ μ (x). Also, let 2 1 2 1 2 f : M −→ M be a map and μ be a fuzzy subset on M . Then we define f (μ) : M −→ [0, 1], 1 2 1 2 as follows: −1 (f (μ))(t) = μ(r),if f (t) =∅ −1 r∈f (t) otherwise we consider (f (μ))(t) = 0. 170 M. NOROUZI AND R. AMERI 2.2. A Connection Between Hypermodules and Fuzzy Hypermodules We recall a connection between fuzzy hypermodules and hypermodules, using the p-cuts of fuzzy sets for p ∈ [0, 1]. By Ref. [11], a structure (M, ⊕, ) is a fuzzy hypermodule over a fuzzy hyperring (R,,) if and only if (M, ⊕ , ) is a hypermodule over the hyperring p p (R, , ), for all p ∈ [0, 1], where p p x ⊕ y ={t ∈ M | (x ⊕ y)(t) ≥ p}, r  s ={u ∈ R | (r  s)(u) ≥ p} p p r x ={z ∈ M | (r x)(z) ≥ p}, r  s ={v ∈ R | (r  s)(v) ≥ p} p p for all x, y ∈ M and r, s ∈ R. Also, according to Refs. [10,11], with every fuzzy hypermodule (M, ⊕, ) over a fuzzy hyperring (R,,), we can associate a hypermodule structure (M, +, ·) over a hyperring (R, , ◦), where x + y ={t ∈ M | (x ⊕ y)(t)> 0}, r s ={u ∈ R | (r  s)(u)> 0} r · x ={z ∈ M | (r x)(z)> 0}, r ◦ s ={v ∈ R | (r  s)(v)> 0} for all x, y ∈ M and r, s ∈ R. On the other hand, let (M, +, ·) be a hypermodule over hyperring (R, , ◦). Consider the following fuzzy hyperoperations for a, b ∈ M and r, s ∈ R a ⊕ b = χ , r a = χ , r  s = χ and r  s = χ . a+b r·a r s r◦s Then (M, ⊕, ) is a fuzzy hypermodule over the fuzzy hyperring (R,,),byRef.[11]. We recall the following theorem regarding connection between subhypermodules and subfuzzy hypermodules and also homomorphism of hypermodules and fuzzy hypermod- ules in Ref. [11]: Theorem 2.1: If (M , ⊕ , ) and (M , ⊕ , ) are fuzzy hypermodules and (M , + , · ) and 1 1 1 2 2 2 1 1 1 (M , + , · ) are the associated hypermodules, then a map f : M −→ M is a homomorphism 2 2 2 1 2 of hypermodules, if f is a homomorphism of fuzzy hypermodules. Theorem 2.2: (1) If N is a subfuzzy hypermodule of (M, ⊕, ) over (R,,),thenNisa subhypermodule of associated hypermodule (M, +, ·) over (R, , ◦). (2) K is a subhypermodule of (M, +, ·) over (R, , ◦) if and only if N is a subhypermodule of associated fuzzy hypermodule (M, ⊕, ) over (R,,). Theorem 2.3: If M and M are hypermodules, then the map f : M −→ M is a homomor- 1 2 1 2 phism of hypermodules if and only if it is a homomorphism of associated fuzzy hypermodules. 2.3. Soft Sets Now, we briefly review some notions concerning soft sets. Let X be an initial universe set and E be a set of parameters. P(X) denotes the power set of X and A ⊆ E. Then, F is called a soft set over X, where F is a mapping given by F : A −→ P(X). In fact, a soft set over X is a parameterised family of subsets of the universe X.For e ∈ A, F(e) may be considered as the set of e-approximate elements of the soft set F . Note to the following example: A FUZZY INFORMATION AND ENGINEERING 171 Example 2.3 ([27]): Consider a soft set F , which describes the attractiveness of houses that one is considering for purchase. Suppose that there are six houses in the universe X,given by X ={h , h , h , h , h , h } and E ={e , e , e , e , e } is a set of decision param- 1 2 3 4 5 6 1 2 3 4 5 eters, where e (i = 1, 2, 3, 4, 5) stand for the parameters ‘expensive’, ‘beautiful’, ‘wooden’, ‘cheap’ and ‘in green surroundings’, respectively. Consider the mapping F by F(e ) ={h , h }, 1 2 4 F(e ) ={h , h }, F(e ) ={h , h , h }, F(e ) ={h , h , h } and F(e ) ={h }. The soft set F is 2 1 3 3 3 4 5 4 1 3 5 5 1 E a parameterised family {F(e ) | 1 ≤ i ≤ 5} of subsets of the set X, and can be viewed as a collection of approximations: F ={expensive houses ={h , h }, beautiful houses ={h , h }, wooden houses ={h , h , E 2 4 1 3 3 4 h }, cheap houses ={h , h , h }, in green surroundings houses ={h }}. 5 1 3 5 1 Example 2.4 ([26]): Consider two universes X and Y and an arbitrary relation, R,from X to Y. Define a set-valued function F : X −→ P(Y) by F(x) ={y ∈ Y | (x, y) ∈ R; x ∈ X}. Obviously, F is a soft set over Y. Definition 2.3: Let F and G be two soft sets over X. G is called a soft subset of F and A B B A denoted by G ⊆ F ,if B ⊆ A and G(x) ⊆ F(x) for each x ∈ B. B A Definition 2.4: (1) Intersection of two soft sets F and G over X such that A ∩ B =∅ is A B the soft set H , where C = A ∩ B and H(c) = F(c) ∩ G(c), for all c ∈ C. This is denoted by F ∩ G = H . A B C (2) Union of two soft sets F and G over X is the soft set H , where C = A ∪ B and for all A B C c ∈ C F(c),if c ∈ A − B H(c) = G(c),if c ∈ B − A F(c) ∪ G(c),if c ∈ A ∩ B. This is denoted by F ∪ G = H . B C (3) If F and G are two soft sets, then F ∧ G is defined as H , where C = A × B and A B A B C H(a, b) = F(a) ∩ G(b), for all (a, b) ∈ C. (4) For two soft sets F and G , F ∨ G , is defined by H , where C = A × B and H(a, b) = A B A B C F(a) ∪ G(b), for all (a, b) ∈ C. For a soft set F , the set Supp(F ) ={x ∈ A | F(x) =∅} is called the support of the soft set A A F .If Supp(F ) =∅, then a soft set F is called non-null. A A A Let F and G be two soft sets over X and X , respectively, and f : X −→ X and g : A −→ B A B be two functions. Then we say that pair (f, g) is a soft function from F to G , denoted by A B (f, g) : F −→ F ,if f (F(x)) = G(g(x)) for all x ∈ A. A B The concept of soft homomorphism on groups was first introduced in Ref. [22]. Consider a soft function (f, g) from F to G . The image of F under the soft function (f, g), A B A denoted by (f, g)(F ) = f (F) , is a soft set over X defined by A B ⎨ f (F(x)),if y ∈ Img f (F)(y) = g(x)=y ∅, otherwise −1 for all y ∈ B. The pre-image of G under the soft function (f, g), denoted by (f, g) (G ) = B B −1 −1 −1 f (G) , is a soft set over X defined by f (G)(x) = f (G(g(x))) for all x ∈ A. It is clear that −1 (f, g)(F ) is a soft subset of G ,and F is a soft subset of (f, g) (G ). A B A B 172 M. NOROUZI AND R. AMERI 3. Soft Hypermodules In this section, we introduce soft hypermodules and investigate their basic properties. Also, soft homomorphisms of hypermodules are discussed, and some illustrative examples are given. Suppose that (M, +, ·) is a hypermodule over a hyperring (R, , ◦) and all soft sets are considered over the hypermodule M. Definition 3.1: Let F be a non-null soft set over a hypermodule M. Then F is called a soft A A hypermodule over M if F(x) is a subhypermodule of M, for all x ∈ Supp(F ). Example 3.1: Consider A = M ={0, 1, 2, 3} and R ={0, 1, 2} with the following hyperoper- ations: + 0123 · 0123 0 {0}{0, 1}{0, 2}{0, 3} 0 {0}{0}{0}{0} 1 {0, 1}{1}{1, 2}{1, 3} 1 {0}{1}{2}{3} 2 {0, 2}{1, 2}{2}{2, 3} 2 {0}{1}{2}{3} 3 {0, 3}{1, 3}{2, 3}{3} 012 ◦ 012 0 {0}{0, 1}{0, 2} 0 {0}{0}{0} 1 {0, 1}{1}{1, 2} 1 {0}{1}{2} 2 {0, 2}{1, 2}{2} 2 {0}{1}{2} Then (M, +, ·) is a hypermodule over hyperring (R, , ◦). Define the set-valued function F : A −→ P(M) by F(x) ={a ∈ M|∃ r ∈ R; a ∈ r · x}. Then F(0) ={0}, F(1) ={0, 1}, F(2) = {0, 2} and F(3) ={0, 3} that of all these are subhypermodules of M. Hence, F is a soft hypermodule over M. Example 3.2: Let (M, +, ·) be the Z-module Z . Suppose that N ={0, 2} which is a sub- module of M.Put, x ⊕ y ={x + y}, r x = rx + N, r s ={r + s} and r ◦ s ={rs} for all x, y ∈ M and r, s ∈ Z. Then, (M, ⊕, ) is a Z-hypermodule constructed by N (see [32,Exam- ple 2.3]). Now, set M = M ∪{a, b, c} such that {a, b, c}∩ M =∅ and consider the following hyperoperations: + 0123 ab c 0 {0}{1}{2}{3}{a}{b}{c} nx +{0, 2}, x ∈ M 1 {1}{2}{3}{0}{a}{b}{c} 2 {2}{3}{0}{1}{a}{b}{c} {a}, x = a and n · x = 3 {3}{0}{1}{2}{a}{b}{c} ⎪ {b}, x = b a {a}{a}{a}{a} M {c}{b} {c}, x = c b {b}{b}{b}{b}{c} M {a} c {c}{c}{c}{c}{b}{a} M for all n ∈ Z. Then (M , + , · ) is a Z-hypermodule by [33, Page 46]. It can be seen that M = M ∪{a}, M = M ∪{b}, M = M ∪{c}, M = M and M = N are the only proper sub- 1 2 3 4 5 hypermodules of M . Now, put I ={0, 1, 2, 3} and similar to Example 3.1, define F(x) = FUZZY INFORMATION AND ENGINEERING 173 {t ∈ M |∃ n ∈ Z; t ∈ n · x} from I to P(M ). Then, F(0) = F(2) = N and F(1) = F(3) = M which are subhypermodules of M . Hence, F is a soft hypermodule over M . Note that if F : J −→ P(M ) for J ={a, b, c}, then F is not a soft hypermodule over M , since F(a) ={a}, F(b) ={b} and F(c) ={c} which are not subhypermodule in M . Notice that the intersection of two subhypermodules of a hypermodule, according to Definition 2.2, is not a subhypermodule, in general. But, if (M, +) is a canonical hyper- group in a hypermodule (M, +, ·), then the intersection of two subhypermodules of M is a subhypermodule. Hence, we have: Theorem 3.1: Let F and G be two soft hypermodules over (M, +, ·) where (M, +) is a canon- A B ical hypergroup, then (1) F ∩ G is a soft hypermodule over M, if it is non-null and A ∩ B =∅. A B (2) If A ∩ B =∅, then F ∪ G is a soft hypermodule over M. A B Proof: (1) Let H = F ∩ G , where C = A ∩ B and H(x) = F(x) ∩ G(x), for all x ∈ C.By C A B hypothesis, Supp(H ) =∅. Since the intersection of subhypermodules of M is a subhy- permodule, H(x) is a subhypermodule of M, for all x ∈ Supp(H ). Hence H is a soft C C hypermodule over M. (2) Since A ∩ B =∅, then c ∈ A ∪ B = C implies that either c ∈ A − B or c ∈ B − A for all c ∈ C.Set H = F ∪ G . Clearly, Supp(H ) =∅. By Definition 2.4, if c ∈ A − B, then H(c) = C A B C F(c) is a subhypermodule of M,and if c ∈ B − A, then H(c) = G(c) is a subhypermodule of M.Thus F ∪ G is a soft subhypermodule over M. A B Corollary 3.1: Let {(F ) |i ∈ I} be a nonempty family of soft hypermodules over (M, +, ·) where (M, +) is a canonical hypergroup, then (1) (F ) is a soft hypermodule over M, if it is non-null and A =∅. i A i i∈I i i∈I (2) If {A |i ∈ I} are pairwise disjoin, then (F ) is a soft hypermodule over M. i i A i∈I i Proof: By using Theorems 3.1, the proof is complete. Theorem 3.2: Suppose that F and G are two soft hypermodules over (M, +, ·) where (M, +) A B is a canonical hypergroup, then (1) F ∧ G is a soft hypermodule over M. A B (2) If F(x) ⊆ G(y) or G(y) ⊆ F(x) for all x ∈ Aand y ∈ B, then F ∨ G is a soft hypermodule A B over M. Proof: (1) Using Definition 2.4, we know that F ∧ G = H , where H(x, y) = F(x) ∩ G(y) A B A×B for all (x, y) ∈ A × B. Obviously, Supp(H ) = Supp(F ) × Supp(G ) =∅. Since the inter- A×B A B section of two subhypermodules is a subhypermodules, it follows that H(x, y) = F(x) ∩ G(y) is a subhypermodule of M for all (x, y) ∈ Supp(H ). Consequently, H is a soft A×B A×B hypermodules over M. 174 M. NOROUZI AND R. AMERI (2) Similar to the proof of (3), by hypothesis, for all (x, y) ∈ Supp(H ) we have H(x, y) = A×B F(x) or H(x, y) = G(y) which are subhypermodules of M. Therefore, H = F ∨ G is a soft A×B A B hypermodule of M. Corollary 3.2: For a nonempty family {(F ) |i ∈ I} of soft hypermodules over (M, +, ·) where (M, +) is a canonical hypergroup, we have (1) (F , A ) is a soft hypermodule over M. i i i∈I (2) If F (a ) ⊆ F (a ) or F (a ) ⊆ F (a ) for all i, j ∈ Iand a ∈ A , then (F , A ) is a soft hyper- i i j j j j i i i i i i i∈I module over M. Proof: By using Theorems 3.2, the proof is complete. Definition 3.2: Let F and G be soft hypermodules over two hypermodules M and M , A B respectively. Let (f, g) be a soft function from F to G .The pair (f, g) is called a homomor- A B phism of soft hypermodules, if f is a homomorphism of hypermodules. Example 3.3: Consider the hypermodules M ={0, 1, 2, 3} and M = R ={0, 1, 2} defined in Example 3.1 (every hyperring is a hypermodule over itself). Let A ={0, 1, 2} and B ={0, 2}. Consider the set-valued functions F : A −→ P(M) and G : B −→ P(M ), which are given by F(x) ={a ∈ M |∃ r ∈ R; a ∈ r · x} and G(x) ={y ∈ M | y ∈ x 0}.Itiseasytoseethat F and G are soft hypermodules over M and M , respectively. Define the function f : M −→ M , which f (0) = 0, f (1) = f (2) = 2and f (3) = 1, then f is a homomorphism of hypermod- ules. Moreover, consider the function g : A −→ B defined by g(0) = 0and g(1) = g(2) = 2. Since f (F(0)) = f ({0}) ={f (0)}={0}= G(0) = G(g(0)), f (F(1)) = f ({0, 1}) ={f (0), f (1)}={0, 2}= G(2) = G(g(1)), f (F(2)) = f ({0, 2}) ={f (0), f (2)}={0, 2}= G(2) = G(g(2)), then (f, g) is a homomorphism of soft hypermodules. Lemma 3.1: Let (f, g) be a soft function from F to G . Then A B (i) If g is onto, then Supp(f (F) ) = g(Supp(F )). B A −1 −1 (ii) Supp(f (G) ) ⊆ g (Supp(G )). A B Proof: (i) Let y ∈ Supp(f (F) ), then ∅ = f (F)(y) = f (F(x)). Hence, there exists x ∈ y=g(x) A such that y = g(x) and F(x) =∅. It implies that y ∈ g(Supp(F )) and so Supp(f (F) ) ⊆ A B g(Supp(F )). On the other hand, g(Supp(F )) ={g(x) | F(x) =∅}. Hence, y ∈ g(Supp(F )) A A A implies that y = g(x) such that F(x) =∅ for x ∈ A. Then f (F)(y) = f (F(x)) =∅ and g(x)=y so y ∈ Supp(f (F) ). Therefore, the proof of (i) is complete. −1 −1 −1 (ii) Suppose that x ∈ Supp(f (G) ). Then, f (G(g(x))) = f (G)(x) =∅. It implies that −1 y = g(x) such that G(y) =∅ for y ∈ B. Hence, x ∈ g (y) and y ∈ Supp(G ). Then, x ∈ −1 g (Supp(G )). B FUZZY INFORMATION AND ENGINEERING 175 Theorem 3.3: Let g : A −→ B be a bijective mapping and (f, g) be a strong homomorphism from F to G , two soft hypermodules over M and M , respectively. Then f (F) is a soft hyper- A B B module over M . Proof: Let y ∈ Supp(f (F) ). By Lemma 3.1 (i), there exists x ∈ Supp(F ) such that y = g(x), B A and F(x) =∅ is a subhypermodule of M. Also, since g is one to one, we have f (F)(y) = f (F(t)) = f (F(x)). y=g(t) Now, since f is a strong homomorphism, f (F(x)) is a subhypermodule of M . It follows that f (F) is a soft hypermodule over M . Theorem 3.4: Let F and G be two soft hypermodules over M and M , respectively, and (f, g) A B −1 a homomorphism from F to G . Then f (G) is a soft hypermodule over M. A B A −1 Proof: Let x ∈ Supp(f (G) ). Then g(x) ∈ Supp(G ) by Lemma 3.1(ii). Hence, G(g(x)) is a A B subhypermodule of M since G is a soft hypermodule. Now, since f is a homomorphism, −1 −1  −1 we get f (G)(x) = f (G(g(x))) is a subhypermodule of M . Therefore, f (G) is a soft hypermodule over M. Theorem 3.5: Let F ,G and H be soft hypermodules over M, M and M , respectively, and A B C (f, g) : F −→ G and (f , g ) : G −→ H be two homomorphisms of soft hypermodules. Then A B B C (f ◦ f, g ◦ g) : F −→ H is a soft (resp. strong) homomorphism of hypermodules. A C Proof: It is straightforward. Definition 3.3: Let F and G be two soft hypermodules over M. Then G is called a soft A B B subhypermodule of F if B ⊆ A and G(x) is a subhypermodule of F(x), for all x ∈ Supp(G ). A B Example 3.4: Consider hypermodule (M, +, ·) over hyperring (R, , ◦) defined in Exam- ple 3.1. Set B ={0, 2} and consider the set-valued function G : B −→ P(M) given by G(x) = {y ∈ M | y ∈ x + 0}. Then G(0) ={0} and G(2) ={0, 2} that are subhypermodules of M and so G is a soft hypermodule over M. Consider the soft hypermodule F defined in Exam- B A ple 3.1. Since B ⊂ A, Supp(G ) ={0, 2}, and clearly G(0) and G(2) are subhypermodules of F(0) and F(2), respectively, then G is a soft subhypermodule of F . B A Theorem 3.6: Let F be a soft hypermodule over (M, +, ·) where (M, +) is a canonical hyper- group, and {(F ) | i ∈ I} be a nonempty family of soft subhypermodules of F . i Ai A (1) (F ) is a soft subhypermodule of F if it is non-null and A =∅. i A A i i∈I i i∈I (2) If {A |i ∈ I} are pairwise disjoin, then (F ) is a soft subhypermodule over M. i i A i∈I i (3) (F ) is a soft subhypermodule over M. i A i∈I i (4) If F (a ) ⊆ F (a ) or F (a ) ⊆ F (a ) for all i, j ∈ Iand a ∈ A , then (F ) is a soft subhy- i i j j j j i i i i i A i∈I i permodule over M. Proof: The proof is similar to the proof of Theorems 3.1 and 3.2.  176 M. NOROUZI AND R. AMERI Theorem 3.7: Let f : M −→ M be a homomorphism of hypermodules. (i) Iffisstrong, andF and G are two soft hypermodules over M such that F is a soft A B A subhypermodule of G , then f (F) is a soft subhypermodule of f (G) . B A B (ii) If F and G are two soft hypermodules over M such that F is a soft subhypermodule of A B A −1  −1 G , then f (F ) is a soft subhypermodule of f (G ) . A B Proof: (i) Clearly, A ⊆ B.Let x ∈ Supp(f (F) ). Hence, f (F(x)) = f (F)(x) =∅ which implies that F(x) =∅. By hypothesis F(x) is a subhypermodule of G(x).Since f is a strong homo- morphism, f (F(x)) is a subhypermodule of f (G(x)) = f (G)(x) and so f (F) is a soft subhy- permodule of f (G) . −1 (ii) Suppose that F is a soft subhypermodule of G ,and x ∈ Supp(f (F )  ). Hence, A B −1  −1 f (F (x)) = f (F )(x) =∅. Then, F (x) =∅ which is a subhypermodule of G (x).Since f −1  −1  −1 is a homomorphism, f (F (x)) is a subhypermodule of f (G (x)) = f (G )(x). It follows −1  −1 that f (F )  is a soft subhypermodule of f (G ) . A B 4. Soft Fuzzy Hypermodules In this section we introduce the notion of a soft fuzzy hypermodule and present some prop- erties of them. In what follow, (M, ⊕, ) is a fuzzy hypermodule over a fuzzy hyperring (R,,) and all soft sets are considered over M. Definition 4.1: A non-null soft set F over a fuzzy hypermodule M is called a soft fuzzy hypermodule if F(x) is a subfuzzy hypermodule of M, for all x ∈ Supp(F ). Example 4.1: Consider the hypermodule (M, +, ·) over the hyperring (R, , ◦) defined in Example 3.1 and A ={0, 1, 2, 3}. Define a ⊕ b = χ , r a = χ , r  s = χ , r  s = χ a+b r·a r s r◦s for all a, b ∈ M and r, s ∈ R. Then (M, ⊕, ) is a fuzzy hypermodule over fuzzy hyperring (R,,),byRef.[11]. Consider the set-valued function F : A −→ P(M) defined by F(x) ={a ∈ M |∃r ∈ R, (r x)(a)> 0}. Then F(0) ={0}, F(1) ={0, 1}, F(2) ={0, 2} and F(3) ={0, 3} that are subfuzzy hypermod- ules of M. Hence, F is a soft fuzzy hypermodule over M. Lemma 4.1: A nonempty subset K is a subfuzzy hypermodule of fuzzy hypermodule (M, ⊕, ) if and only if K is a subhypermodule of (M, ⊕ , ) for all p ∈ [0, 1]. p p Proof: K is a subfuzzy hypermodule of (M, ⊕, ) iff (K, ⊕, ) is a fuzzy hypermodule iff (K, ⊕ , ) for all p ∈ [0, 1] is a hypermodule iff K is a subhypermodule of (M, ⊕ , ) for p p p p all p ∈ [0, 1]. Lemma 4.2: Let N and K be two subfuzzy hypermodules of (M, ⊕, ) where (M, ⊕) is a canonical fuzzy hypergroup. Then N ∩ K is a subfuzzy hypermodule of M. FUZZY INFORMATION AND ENGINEERING 177 Proof: Let x, y ∈ N ∩ K, t ∈ M and (x ⊕ y)(t)> 0. Since, x, y ∈ N, x, y ∈ K,and N and K are subfuzzy hypermodules, then t ∈ N and t ∈ K,and so t ∈ N ∩ K. Similarly, (r x)(t)> 0 implies that t ∈ N ∩ K for all r ∈ R. Also, for z ∈ N ∩ K and p ∈ (0, 1], by associated hyper- module (M, ⊕ , ),wehave p p z ⊕ (N ∩ K) (t) ≥ p ⇐⇒ t ∈ z ⊕ (N ∩ K) ⇐⇒ t ∈ (z ⊕ N) ∩ (z ⊕ K) p p ⇐⇒ t ∈ N ∩ K (by Lemma 4.1) ⇐⇒ χ (t) ≥ p. N∩K Now, consider the associated hypermodule (M, +, ·). Then z ⊕ (N ∩ K) (t) = 0 ⇔ t ∈ z + (N ∩ K) = (z + N) ∩ (z + K) = N ∩ K ⇔ χ (t) = 0. N∩K It follows that z ⊕ (N ∩ K) = χ . Therefore, the proof is complete. N∩K Theorem 4.1: Let F and G be two soft fuzzy hypermodules over M where (M, ⊕) is a canon- ical fuzzy hypergroup. Then: (1) F ∩ G is a soft fuzzy hypermodule over M, if it is non-null and A ∩ B =∅. A B (2) If A ∩ B =∅, then F ∪ G is a soft fuzzy hypermodule over M. A B (3) F ∧ G is a soft fuzzy hypermodule over M. A B (4) If F(x) ⊆ G(y) or G(y) ⊆ F(x) for all x ∈ Aandy ∈ B, then F ∨ G is a soft fuzzy hypermod- A B ule over M. Proof: By using Lemma 4.2, similar to the proof of Theorems 3.1 and 3.2, the proof is completed. Said results in Corollaries 3.1 and 3.2 are valid for soft fuzzy hypermodules, similarly. Definition 4.2: A soft fuzzy hypermodule G is said to be a soft subfuzzy hypermodule of soft fuzzy hypermodule F ,if B ⊆ A and G(x) is a subfuzzy hypermodule of F(x), for all x ∈ Supp(G ). Example 4.2: Set B ={0, 2} and consider the set-valued function G : B −→ P(M) given by G(x) ={y ∈ M | (x ⊕ 0)(y)> 0} over the fuzzy hypermodule (M, ⊕, ) and soft fuzzy hypermodule F defined in Example 4.1. Then G(0) ={0} and G(2) ={0, 2} which are sub- fuzzy hypermodules of M and also F(0) and F(2), respectively. Hence G is a soft subfuzzy hypermodule of F . Definition 4.3: The pair (f, g) is called a homomorphism of soft fuzzy hypermodules F and G over two fuzzy hypermodules M and M , respectively, if (f, g) is a soft function from F to B A G , then f is a homomorphism of fuzzy hypermodules. Example 4.3: Consider two fuzzy hypermodules (M, ⊕, ) and (M = R,,) as in Exam- ple 4.1. Set A ={0, 1, 2}, B ={0, 2} and let set-valued functions F : A −→ P(M) and G : 178 M. NOROUZI AND R. AMERI B −→ P(M ), which are given by F(x) ={a ∈ M |∃ r ∈ R; (r x)(a)> 0} and G(x) ={y ∈ M | (x  0)(y)> 0}. Then F and G are soft fuzzy hypermodules over M and M ,respec- A B tively. Let f and g as in Example 3.3. By connection of hypermodules and fuzzy hypermod- ules, it concludes that f is a homomorphism of fuzzy hypermodules. This implies that, (f, g) is a homomorphism of soft fuzzy hypermodules. Lemma 4.3: Let f : M −→ M be a homomorphism of fuzzy hypermodules and N and K sub- fuzzy hypermodules of M and M , respectively. (1) If fisstrong, thenf (N) is a subfuzzy hypermodule of M . −1 (2) f (K) is a subfuzzy hypermodule of M. Proof: (1) Suppose that f (x), f (y) ∈ f (N), t ∈ M and (f (x) ⊕ f (y))(t)> 0. Consider the associated hypermodule (M, +, ·). By Theorem 2.2 (1) and Theorem 2.1, it implies that t ∈ f (x) + f (y) = f (x + y) ⊆ f (N). Also, similar to the proof of Lemma 4.2, we can show that f (x) ⊕ f (N) = χ ,and (r f (x))(t)> 0 implies that t ∈ f (N) for r ∈ R. Similarly, (2) will be f (N) proved. By Lemma 4.3, it can be seen that Theorems 3.3, 3.4, 3.5 and 3.7 are valid for soft fuzzy hypermodules similarly. 5. Connections Between Soft Hypermodules and Soft Fuzzy Hypermodules In this section, the connections between soft hypermodules and soft fuzzy hypermodules and also their homomorphisms are investigated based on associated hyperoperations and fuzzy hyperoperations ([11]). The connections between fuzzy hyperoperations and associated hyperoperations have been considered by Sen, Ameri and Chowdhury in the context of semihypergroups and hypergroups ([9]), by Leoreanu-Fotea and Davvaz in hyperrings ([10]), and by Leoreanu- Fotea in hypermodules ([11]). The following theorems establish a similar result for soft hypermodules and soft fuzzy hypermodules. Theorem 5.1: Let F be a soft fuzzy hypermodule over fuzzy hypermodule (M, ⊕, ) on a fuzzy hyperring (R,,). Then F is a soft hypermodule over associated hypermodule (M, +, ·) on associated hyperring (R, , ◦). Proof: Since F is soft fuzzy hypermodule, F(x) is a subfuzzy hypermodule of (M, ⊕, ) for all x ∈ A. By Theorem 2.2 (1), F(x) is a subhypermodule of associated hypermodule M,for all x ∈ A. Therefore, F is a soft hypermodule over (M, +, ·). If we denote by SHM the class of all soft hypermodules and by SFHM the class of all soft fuzzy hypermodules, then we can consider the map  : SFHM −→ SHM such that  ((F , ⊕, )) = (F , +, ·). S A A FUZZY INFORMATION AND ENGINEERING 179 Example 5.1: Consider soft fuzzy hypermodule F over fuzzy hypermodule (M, ⊕, ) defined in Example 4.1. We have F(x) ={a ∈ M |∃r ∈ R; (r x)(a)> 0}={a ∈ M |∃r ∈ R; a ∈ r · x} for all x ∈ A. F is a soft hypermodule over associated hypermodule (M, +, ·) by Example 3.1. Lemma 5.1: Let (M, +, ·) be a hypermodule on (R, , ◦),and (M, ⊕, ) the associated fuzzy hypermodule on associated fuzzy hyperring (R,,). Then a ⊕ N = χ for every subhyper- a+N module N of M and for all a ∈ N. Proof: Let p ∈ (0, 1] and t ∈ M. Then (a ⊕ N)(t) ≥ p ⇔ (a ⊕ n)(t) ≥ p ⇔∃b ∈ N; χ (t) ≥ p ⇔ t ∈ a + N = N a+b n∈N ⇔ χ (t) ≥ p. Also, (a ⊕ N)(t) = 0iff t ∈ a + N = N iff χ (t) = 0. Therefore, for all p ∈ [0, 1] we have (a ⊕ N)(t) ≥ p ⇐⇒ χ (t) ≥ p. Hence, a ⊕ N = χ . N a+N The next theorem shows that we can obtain a soft fuzzy hypermodule by a soft hyper- module. Theorem 5.2: F is a soft hypermodule over a hypermodule (M, +, ·) on a hyperring (R, , ◦) if and only if F is a soft fuzzy hypermodule over an associated fuzzy hypermodule (M, ⊕, ) on associated fuzzy hyperring (R,,). Proof: By Theorem 2.2 (2), we have F is a soft hypermodule over M if and only if F(x) is a subhypermodule of M for all x ∈ A if and only if F(x) is a subfuzzy hypermodule of asso- ciated fuzzy hypermodule M for all x ∈ A if and only if F is a soft fuzzy hypermodule over associated fuzzy hypermodule M. Hence, there exists a map  : SHM −→ SFHM such that  ((F , +, ·)) = (F , ⊕, ). S S A A Example 5.2: Consider soft hypermodule G over hypermodule (M, +, ·) defined in Exam- ple 3.4. By associated fuzzy hyperoperations we have G(x) ={y ∈ M | y ∈ x + 0}={y ∈ M | χ (y) = 1}={y ∈ M | (x ⊕ 0)(y)> 0}. x+0 G is a soft fuzzy hypermodule over associated fuzzy hypermodule (M, ⊕, ) (Example 4.1). Now, we investigate the connection between homomorphism of soft hypermodules and homomorphism of soft fuzzy hypermodule. Theorem 5.3: Let (F , ⊕ , ) and (G , ⊕ , ) be two soft fuzzy hypermodules over A 1 1 B 2 2 fuzzy hypermodules (M , ⊕ , ) and (M , ⊕ , ), respectively, and also (F , + , · ) = 1 1 1 2 2 2 A 1 1 ((F , ⊕ , )) and (G , + , · ) =  ((G , ⊕ , )) be associated soft hypermodules. If S A 1 1 B 2 2 S B 2 2 180 M. NOROUZI AND R. AMERI (f, g) is a homomorphism of soft fuzzy hypermodules, then (f, g) is a homomorphism of soft hypermodule homomorphism, too. Proof: By Theorem 2.1, if f is a homomorphism of fuzzy hypermodules, then f is a homo- morphism of associated hypermodules, too. This completes the proof. Theorem 5.4: Let (F , + , · ) and (G , + , · ) be two soft hypermodules over hypermodules A 1 1 B 2 2 M and M and (F , ⊕ , ) =  ((F , + , · )) and (G , ⊕ , ) =  ((G , + , · )) be asso- 1 2 A 1 1 S A 1 1 B 2 2 S B 2 2 ciated soft fuzzy hypermodules. If (f, g) is a homomorphism of soft fuzzy hypermodule if and only if (f, g) is a homomorphism of soft hypermodules. Proof: By Theorem 2.3, the proof is complete. 6. Conclusion and Future Work We combined three theories soft sets, fuzzy sets and algebraic hyperstructures (here hyper- modules) to introduce soft fuzzy hypermodules, as a generalisation of soft fuzzy modules. We studied some basic properties of classes soft hypermodules and soft fuzzy hyper- modules. Furthermore, we introduced associated (fuzzy) hyperoperations and soft (fuzzy) homomorphisms to establish a connection between the class of soft hypermodules and soft fuzzy hypermodules. This paper provided the general tools to study soft set theory and soft fuzzy concept on algebraic hyperstructures ; therefore, it can be considered as a generalisa- tion of the works in this area (such as Refs. [26–29]). Also note that the presented results in this paper, can be easily extended for (m, n)-hyperstructures such as (m, n)-hypermodules or e-hyperstructures. Disclosure statement No potential conflict of interest was reported by the authors. Notes on Contributors Morteza Norouzi is an assistance professor in Department of Mathematics at University of Bojnord, Bojnord, Iran. He received BSc, MSc, and PhD degrees in Pure Mathematics (Algebra, Algebraic Hyperstructures) from University of Mazandaran, Babolsar, Iran (2004–2014). His research interests is in connection with algebraic hyperstructures: fundamental relations, (m, n)-ary hyperstructures, chemical hyperstructures and fuzzy algebraic hyperstructures. Reza Ameri is a professor in Department of Mathematics, School of Mathematics, Statistics and Com- puter Science, University of Tehran, Tehran, Iran. He received his M.S. in 1993 in Fuzzy Algebra, and his Ph.D. in 1997 in algebra(Algebraic Hyperstructures), from Shahid Bahonar University of Kerman, Kerman, Iran. He spent one year as visiting professor in Montreal University in Canada in 2004. His research interest is Algebra and Algebraic Hyperstructures: Category Theory, Universal Algebra and Fuzzy Mathematics. ORCID M. Norouzi http://orcid.org/0000-0001-9850-1126 FUZZY INFORMATION AND ENGINEERING 181 References iem [1] Marty F. Sur une generalization de la notion de groupe. In: 8 congres des Mathematiciens Scandinaves. Stockholm. 1934. p. 45–49. [2] Corsini P. Prolegomena of hypergroup theory. 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Journal

Fuzzy Information and Engineering – Taylor & Francis

Published: Apr 3, 2022

Keywords: Soft set; soft hypermodule; soft fuzzy hyermodule; 20N20; 08A72

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