Abstract:
Fuzzy sets, rough sets, and later on IF sets became useful mathematical tools for solving various decision making problems and data mining problems. Molodtsov introduced another concept soft set theory as a general frame work for reasoning about vague concepts. Since most of the data collected are either linguistic variable or consist of vague concepts so IF set and soft set help a lot in data mining problem. The aim of this paper is to introduce the concept of IF soft lower rough approximation and IF upper rough set approximation. Also, some properties of this set are studied, and also some problems of decision making are cited where this concept may help. Further research will be needed to apply this concept fully in the decision making and data mining problems.

Abstract:
The notion of an $n$-ary group is a natural generalization of the notion of a group and has many applications in different branches. In this paper, the notion of (normal) fuzzy $n$-ary subgroup of an $n$-ary group is introduced and some related properties are investigated. Characterizations of fuzzy $n$-ary subgroups are given.

Abstract:
The notion of intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. In this paper, we apply the concept of intuitionistic fuzzy sets to Hv-rings. We introduce the notion of an intuitionistic fuzzy Hv-ideal of an Hv-ring and then some related properties are investigated. We state some characterizations of intuitionistic fuzzy Hv-ideals. Also we investigate some natural equivalence relations on the set of all intuitionistic fuzzy Hv-ideals of an Hv-ring.

Abstract:
Based on the works of Axtell et al., Anderson et al., and Ghanem on associate, domainlike, and presimplifiable rings, we introduce new hyperrings called associate, hyperdomainlike, and presimplifiable hyperrings. Some elementary properties of these new hyperrings and their relationships are presented. 1. Introduction The study of strongly associate rings began with Kaplansky in [1] and was further studied in [2–5]. Domainlike rings and their properties were presented by Axtell et al. in [6]. Presimplifiable rings were introduced by Bouvier in the series of papers [7–11] and were later studied in [2–4]. Further properties of associate and presimplifiable rings were recently presented by Ghanem in [12]. The theory of hyperstructures was introduced in 1934 by Marty [13] at the 8th Congress of Scandinavian Mathematicians. Introduction of the theory has caught the attention and interest of many mathematicians and the theory is now spreading like wild fire. The notion of canonical hypergroups was introduced by Mittas [14]. Some further contributions to the theory can be found in [15–19]. Hyperrings are essentially rings with approximately modified axioms. Hyperrings are of different types introduced by different researchers. Krasner [20] introduced a type of hyperring where + is a hyperoperation and is an ordinary binary operation. Such a hyperring is called a Krasner hyperring. Rota in [21] introduced a type of hyperring where + is an ordinary binary operation and is a hyperoperation. Such a hyperring is called a multiplicative hyperring. de Salvo [22] introduced and studied a type of hyperring where + and are hyperoperations. The most comprehensive reference for hyperrings is Davvaz and Leoreanu-Fotea’s book [18]. Some other references are [23–31]. In this paper, we present and study associate, hyperdomainlike, and presimplifiable hyperrings. The relationships between these new hyperrings are presented. 2. Preliminaries In this section, we will provide some definitions that will be used in the sequel. For full details about associate, domainlike, and presimplifiable rings, the reader should see [1, 4–6, 12]. Also, for details about hyperstructures and hyperrings, the reader should see [12]. Definition 1. Let be a commutative ring with unity. (1) is called an associate ring if whenever any two elements generate the same principal ideal of , there is a unit such that .(2) is called a domainlike ring if all zero divisors of are nilpotent.(3) is called a presimplifiable ring if, for any two elements with , we have or .(4) is called a superassociate ring if

Abstract:
Hypergroups first were introduced by Marty in 1934. Up to now many researchers have been working on this field of modern algebra and developed it. It is purpose of this paper to provide examples of hypergroups associated with chemistry. The examples presented are connected to construction from chain reactions.

Abstract:
We have introduced a new nonassociative class of Abel-Grassmann's groupoid, namely, intraregular and characterized it in terms of its (∈,∈∨)-fuzzy quasi-ideals.

Abstract:
This paper deals with the algebraic hypersystems. The notion of regularity of different type of algebraic systems has been introduced and characterized by different authors such as Iseki, Kovacs, and Lajos. We generalize this notion to algebraic hypersystems giving a unified generalization of the characterizations of Kovacs, Iseki, and Lajos. We generalize also the concept of ideal introducing the notion of -hyperideal and hyperideal of an algebraic hypersystem. It turns out that the description of regularity in terms of hyperideals is intrinsic to associative hyperoperations in general. The main theorem generalizes to algebraic hypersystems some results on regular semigroups and regular rings and expresses a necessary and sufficient condition by means of principal hyperideals. Furthermore, two more theorems are obtained: one is concerned with a necessary and sufficient condition for an associative, commutative algebraic hypersystem to be regular and the other is concerned with nilpotent elements in the algebraic hypersystem. 1. Introduction Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, and coding theory. Hyperstructure theory was introduced in 1934, when Marty [1] defined hypergroups based on the notion of hyperoperation, began to analyze their properties, and applied them to groups. In the following decades and nowadays, a number of different hyperstructures are widely studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics by many mathematicians. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Several books have been written on hyperstructure theory; see [2–5]. A recent book on hyperstructures [3] points out their applications in rough set theory, cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs, and hypergraphs. Another book [4] is devoted especially to the study of hyperring theory. Several kinds of hyperrings are introduced and analyzed. The volume ends with an outline of applications in chemistry and physics, analyzing several special kinds of hyperstructures: -hyperstructures and transposition hypergroups. The theory of suitable modified hyperstructures can serve as a mathematical background in the field of quantum communication systems. -ary generalizations of algebraic

Abstract:
The notion of intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. In this paper, we consider the intuitionistic fuzzification of the concept of sub-hyperquasigroups in a hyperquasigroup and investigate some properties of such sub-hyperquasigroups. In particular, we investigate some natural equivalence relations on the set of all intuitionistic fuzzy sub-hyperquasigroups of a hyperquasigroup.