Soft Neutrosophic Modules

by mathematics (moPt, Article Soft Neutrosophic Modules Mikail Bal and Necati Olgun *® Department of Mathematics, Gaziantep University, Gaziantep 27310, Turkey; mikailbal46@hotmail.com * Correspondence: olgun@gantep.edu.tr check for Received: 22 October 2018; Accepted: 8 December 2018; Published: 12 December 2018 iS) updates Abstract: In this study, for the first time, we study some basic definitions of soft neutrosophic modules in algebra being generalized and its several related properties, structural characteristics are investigated with suitable examples. In this paper, we utilized neutrosophic soft sets and neutrosophic modules. As a result, we defined soft neutrosophic modules. After weak soft neutrosophic modules and strong soft neutrosophic modules are described and illustrated by examples. Finally soft neutrosophic module homomorphism is defined and soft neutrosophic module isomorphism is explained. Keywords: strong soft neutrosophic module; weak soft neutrosophic module; soft neutrosophic submodule; soft neutrosophic ring 1. Introduction In 1980, Florentin Smarandanche characterized the notion of neutrosophy as a new branch of philosophy. “It is a branch of philosophy which studies the nature, scope and origin of neutralities, as well as their interactions with different ideational spectra”. It is the base of neutrosophic logic that is an extension fuzzy logic in that indeterminacy is inclusive. He introduced the concept of (T-truth, I-indeterminacy, and F-falsity) memberships. Neutrosophic set is a power general formal structure which generalizes the concept of the intuitionistic fuzzy set, interval valued fuzzy set, fuzzy set, classic set etc. Struggle with uncertainties is a big problem in many areas like medical science, environment, economics, social sciences, etc. These kinds of problems cannot be overcome by classical methods, because there are inherent difficulties in the classical methods. The reason for these difficulties may be due to the inadequacy of the theories of parameterization tools. For Dealing with these kinds of difficulties, Molodtsov suggested a new mathematical tool for dealing with uncertainties. This is called soft set theory [1]. The algebraic structure of set theories dealing with uncertainties has also been studied by some authors. Smarandanche and others, studied neutrosophic algebraic structures [2-4] by using the idea of neutrosophic theory and this was the first neutrosofication of algebraic structures. Some authors [2-21] have studied neutrosophic algebraic structures and nowadays, researchers are still interested in this subject. The importance of this neutrosophic structures continues to increase. Recently, researchers have done a number of soft neutrosophic structures formed by combining soft structures and neutrosophic structures [18,20,21]. Therefore, we decided to investigate the effects of this subject on modules of algebraic structures. Modules are basic structures in algebraic structures and are generalization of the vector spaces. This topic has not been studied yet. In this paper, for the first time we introduce ‘Soft Neutrosophic Module’, ‘Weak Soft-N Module’, and Strong Soft-N Module’. Finally, we described the soft-N module homomorphism and the soft-N module isomorphism in module theory, using neutrosophic soft sets. This is new topic in the module Mathematics 2018, 6, 323; doi:10.3390/math6120323 www.mdpi.com/journal/mathematics Mathematics 2018, 6, 323 2 of 8 theory and neutrosophic logic. The rest of the paper is organized as follows. After the literature review in Section 1 and Preliminaries in Section 2, we introduce soft neutrosophic module, weak soft-N module and strong soft-N module and also examine the relations between weak soft-N module and strong soft-N module in Section 3. Finally, conclusions are given in Section 4. 2. Preliminaries In this chapter, we remember some basic definition for soft sets and neutrosophic modules. These are related to fuzzy sets, soft sets, neutrosophic sets which are found in the literature [1,2,5,22—26]. 2.1. Soft Sets Theory Let U be an initial universe set and E be a set of parameters. The power set of U is denoted by P(U) and A C E. Molodtsov defined the notation of a set in the following way: Definition 1. A pair (F, A) is called a soft set over U, where F is a mapping given by: F:A— P(U) [1] Definition 2. For two soft sets (F, A) and (G, B) over a general universe U, we say that (F, A) is a soft subset of (G, B) if. (i) AC B,and (ii) F(x) and G(x) are identical approximations V x € A. We write (F, A) C (G,B). (E, A) is said to be a soft super set of (G, B) if (G, B) is a soft subset of (F, A). We denote it by (FE, A) D (G, B) (see [1]). Definition 3. Two soft sets (F, A) and (G, B) on a common universe U are said to be soft equal if (F, A) is a soft subset of (G, B) and (G, B) is a soft subset of (FE, A) (see [1]). Definition 4. Union of two soft sets of (F, A) and (G, B) over the common universe U is th