International Journal of Mathematics and Mathematical Sciences

Volume 2018, Article ID 5712676, 6 pages

https://doi.org/10.1155/2018/5712676

## New Branch of Intuitionistic Fuzzification in Algebras with Their Applications

Department of Mathematics, College of Science, Basrah University, Basrah 61004, Iraq

Correspondence should be addressed to Shuker Mahmood Khalil; moc.liamg@melasla.rekuhs

Received 28 February 2018; Revised 22 May 2018; Accepted 11 June 2018; Published 10 July 2018

Academic Editor: Susana Montes

Copyright © 2018 Samaher Adnan Abdul-Ghani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The intuitionistic fuzzification in algebras about the concepts of ideals and subalgebras given with several related characterizations is considered. Some new concepts like intuitionistic fuzzy ideal (), intuitionistic fuzzy subalgebra (), homomorphism, and intuitionistic fuzzy ideal () are introduced and some of their descriptions are given in this work. Further, we show some applications on the family of all intuitionistic fuzzy subalgebras in algebra like the binary relations , and on . Also, their equivalence classes are given and studied.

#### 1. Introduction

The fuzzy set (FS) as suggested by Zadeh [1] in 1965 is a regulation to vagueness and encounter uncertainty. A FS maps each element of the universe of discourse to the interval After the introduction of fuzzy sets theory by him, many mathematicians were conducted on the generalizations of the this concept and studied in the groups, algebras, and soft spaces (see [2–5]). By including a fuzzy set the degree of nonmembership, Atanassov [6] in 1986 suggested the intuitionistic fuzzy set (IFS), which seems more precise for provides opportunities and uncertainty quantification to accurately model a problem based on existing knowledge and monitoring. Also, this notion is discussed in different fields (see [7–11])

algebra, class of algebra of logic, was investigated by Imai and Iseki [12]. After that, the notion of algebras was investigated by Neggers and Kim [13]. In 2017, the concepts of algebra, ideal, ideal, subalgebra, and permutation topological algebra were first proposed by Mahmood and Abud Alradha [14]. Next, they showed the notion of the soft algebra and soft edge algebra [15].

In the present work, the notions of intuitionistic fuzzy ideal (), intuitionistic fuzzy subalgebra (), homomorphism, and intuitionistic fuzzy ideal () are introduced. Further, we show some applications on the family of all intuitionistic fuzzy subalgebras in algebra like the binary relations , and on . Also, their equivalence classes are given and studied.

#### 2. Preliminaries and Notations

We will recall basic definitions and results to obtain properties developed in this work.

*Definition 1 (see [16]). *An intuitionistic fuzzy set (IFS, in short) over the universe is defined by , where : [0; 1], : [0; 1] with 0 , . and are real numbers and their values represent the degree of membership and nonmembership of to , respectively.

*Definition 2 (see [6]). *The IF whole and empty sets of are defined by and , respectively.

##### 2.1. Basic Relations and Operations on Intuitionistic Fuzzy Sets [7]

Assume and are two IF sets of . We deduced the following relations:(1)[inclusion] iff and , ,(2)[equality] iff and ,(3)[intersection] , (4)[union] , ,(5)[complement] .

*Definition 3 (see [14]). *We say is algebra if () is a binary operation on with a constant and such that(1),(2),(3) imply that ,(4)For all imply that .

*Definition 4 (see [14]). *Assume is a algebra and . We say is a subalgebra of if , .

*Definition 5 (see [14]). *Assume is algebra and . We say is ideal of if (1) imply ,(2) and imply , .

*Definition 6 (see [14]). *Assume is a algebra and subset of . We say is a ideal of if (1),(2) and , *.*

*Definition 7 (see [11]). *Assume that is an IFS in and The set (resp., is said to be *level **cut *(resp., *level **cut*) of .

#### 3. Intuitionistic Fuzzy Subalgebras in Algebras

In this section, we introduce some new concepts, such as (), (), (), and homomorphism which are introduced and discussed. Further, some binary relations , and on are given, and some basic properties are shown.

*Definition 8. *Assume is a algebra and is IFS of . We say is an () of if and , .

*Example 9. *Let be algebra with Table 1.

Then, is an () of .