MV-Modules in View of soft Set Theory

Çankaya University Journal of Science and Engineering Volume 13, No. 1 (2016) 001–015 MV -Modules in View of Soft Set Theory A. Erami1,∗ , A. Hasankhani2 and A. Borumand Saeid 3 1 Department of Mathematics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran 2 Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran 3 Department of Pure Mathematics, Shahid Bahonar University of Kerman Kerman, Iran. ∗ Corresponding author: Abstract: In this paper, the concept of a soft MV -module is introduced and some examples are provided. Then, different types of intersections and unions of the family of soft MV -modules are established. Moreover, the notions of soft MV-submodules and soft MV -module homomorphisms are introduced and some of their properties are studied. Keywords: Soft MV -module, Soft set, MV -module. 1. Introduction MV -algebra was introduced by Chang [6] in 1958 as the algebraic structure corresponding to the infinitely-many-valued Lukasiewicz logic. Since then, this structure has been developed from an algebraic point of view by many mathematicians. In 1986, Mundici proved that the category of MV -algebras is equivalent to the category of abelian l-groups with strong unit (see [7]). Noje and Bede [25] introduced the concept of vectorial MV -algebra and showed that the RGB model (the color model of a pixel on the screen) has the vectorial MV -algebra structure. Di Nola et al. [9] introduced the notion of an MV -module over a PMV -algebra and established that, for a fixed lu-ring (R, ν ), the category of lu-modules over (R, ν ) is equivalent to the category of MV -modules over Γ(R, ν ). Forouzesh et al. [13] introduced the notion of prime A-ideals in MV modules and studied about annihilators of A-ideals. They proved that, if h : M → N is an MV module homomorphism, then all prime A-ideals of N and prime A-ideals of M that contain ker(h) are in a one to one correspondence. On the other hand, soft set theory was initiated by Molodtsov [23] in 1999 as a new mathematical tool for modeling the uncertainties arising from the parametrization of elements of a universe. He mentioned several directions for the applications of soft sets. In fact, before soft set theory, there have been some mathematical theories such as probability theory, fuzzy set theory, rough set theory, vague set theory, and interval mathematics theory for dealing with uncertainties. However, the superiority of the soft set theory compared with other mathematical tools, is its ability of ISSN 1309 - 6788 c 2016 Çankaya University 2 A. Erami et al. parametrization. Maji et al. [21] studied several operations on the theory of soft sets. Some authors have discussed the applications of (fuzzy) soft sets in decision making problems (see [5, 22]). Aktas and Cagman [3] compared soft sets to the related concepts of fuzzy sets and rough sets. They also introduced the notion of soft groups. After them, soft algebraic structures have been studied by many authors (see [1, 10, 14, 16, 19, 27, 29]). Jun [17] introduced and investigated soft BCK/BCI-algebras. Feng et al. [12] applied soft set theory to the study of semirings and initiated the notion called a soft semiring. Afkhami et al. [2] presented the concept of a soft nexus. Zhu [30] introduced the concept of soft BL-algebras. Moreover, by combination of fuzzy set theory with soft set theory, fuzzy soft algebraic structures were born. For example, Hadipour et al. [14] defined the notion of fuzzy soft BF-algebra and investigated the level subset, union and intersection, fuzzy soft image and fuzzy soft inverse image of them. Murali [24] introduced the concept of a fuzzy soft Γ-semiring and fuzzy soft k-ideal over a Γ-semiring and studied some of their algebraical properties. Ersoy et al. [11] introduced the concept of an idealistic fuzzy soft Γ-near-ring and derived some results on this structure. The most soft algebraic structures are defined as follows: for a set of parameters E and a general algebra X , a pair (F, E) is called a soft general algebra over X if F is a mapping of E into the set of all subsets of the set X such that for each e ∈ E, F(e) is the empty set or a subalgebra of X . In this paper, at first, some definitions and results related to soft set and MV -modules is reviewed. Then, the notion of a soft MV -module is introduced and some examples are provided. Two examples of MV -modules that have no proper submodules are given in Section 3. Then, the extended intersection, restricted intersection, V -intersection, extended union, restricted union and ∨-union of the family of soft MV -modules are established. Moreover, the notions of soft MV -module homomorphisms, soft isomorphic MV -modules and soft MV -submodules are introduced and some of their properties are studied. Also, it is shown that there is a one-to-one correspondence between the soft MV-submodules of two soft isomorphic MV-modules. 2. Preliminaries Some definitions and results about soft set and MV -module are presented in this section. Let U be an initial universe set and let E be a set of parameters. Molodtsov [23] defined the soft set in the following way: Definition 1. A pair (F, E) is called a soft set (over U ) if F is a mapping of E into the set of all subsets of the set U . CUJSE 13, No. 1 (2016) MV -Modules in View of Soft Set Theory 3 Definition 2. [26] Let (F, A) and (G, B) be two soft sets over U . Then, (i) (F, A) is said to be a soft subset of (G, B), denoted by (F, A) ⊆ (G, B), if A ⊆ B and F(a) ⊆ G(a) for all a ∈ A, (ii) (F, A) and (G, B) are said to be soft equal, denoted by (F, A) = (G, B), if (F, A) ⊆ (G, B) and (G, B) ⊆ (F, A). The next definition introduces three types of intersections and three types of unions of the family of soft sets over a common universe set: Definition 3. [28] For a family {(Fi , Ai ) | i ∈ I } of soft sets over U , we give some definitions as follows: • The extended intersection of the family (Fi , Ai ) is defined as the soft set T e i∈I (Fi , Ai ) = (H,C), where C = S i∈I Ai and H(x) = T i∈I(x) Fi (x) where I(x) = {i ∈ I | x ∈ Ai } for all x ∈ C. • The restricted intersection of the family (Fi , Ai ) is defined as the soft set where C = T i∈I Ai and H(x) = T e i∈I (Fi , Ai ) = (H,C), ⊓ i∈I Fi (x) for all x ∈ C. • The extended union of the family (Fi , Ai ) is defined as the soft set S e i∈I (Fi , Ai ) = (H,C), where C = S i∈I Ai , H(x) = S i∈I(x) Fi (x) and I(x) = {i ∈ I | x ∈ Ai } for all x ∈ C. • The restricted union of the family (Fi , Ai ) is defined as the soft set F e i∈I (Fi , Ai ) = (H,C), where C = T / and H(x) = i∈I Ai 6= 0 S i∈I Fi (x) for all x ∈ C. • The ∧-intersection of the family (Fi , Ai ) is defined as the soft set V e i∈I (Fi , Ai ) = (H,C), where C = ∏i∈I Ai and H((ai )i∈I ) = T i∈I Fi (ai ) for all (ai )i∈I ∈ C. • The ∨-union of the family (Fi , Ai ) is defined as the soft set W e i∈I (Fi , Ai ) = (H,C), where C = ∏i∈I Ai and H((ai )i∈I ) (...truncated)