Some Properties of Intuitionistic Fuzzy Soft Rings

Abstract

Maji et al. introduced the concept of intuitionistic fuzzy soft sets, which is an extension of soft sets and intuitionistic fuzzy sets. In this paper, we apply the concept of intuitionistic fuzzy soft sets to rings. The concept of intuitionistic fuzzy soft rings is introduced and some basic properties of intuitionistic fuzzy soft rings are given. Intersection, union, AND, and OR operations of intuitionistic fuzzy soft rings are defined. Then, the deffinitions of intuitionistic fuzzy soft ideals are proposed and some related results are considered.

1. Introduction

Uncertain or imprecise data are inherent and pervasive in many important applications in the areas such as economics, engineering, environment, social science, medical science, and business management. Uncertain data in those applications could be caused by data randomness, information incompleteness, limitations of measuring instrument, delayed data updates, and so forth. Due to the importance of those applications and the rapidly increasing amount of uncertain data collected and accumulated, research on effective and efficient techniques that are dedicated to modeling uncertain data and tackling uncertainties has attracted much interest in recent years and yet remained challenging at large. There have been a great amount of research and applications in the literature concerning some special tools like probability theory, (intuitionistic) fuzzy set theory, rough set theory, vague set theory, and interval mathematics. However, all of these have theirs advantages as well as inherent limitations in dealing with uncertainties. One major problem shared by those theories is their incompatibility with the parameterizations tools. Soft set theory [1] was firstly proposed by a Russian researcher, Molodtsov, in 1999 to overcome these difficulties.

At present, work on the extension of soft set theory is progressing rapidly. Maji et al. proposed the concept of fuzzy soft set [2] and then gave its application. Roy and Maji presented a method of object recognition from an imprecise multiobserver data [3]. Yao et al. proposed the concept of fuzzy sets and defined some operations on fuzzy soft sets [4].

From the above discussion, we can see that all of these works are based on Zadeh’s fuzzy sets theory which was generalized to intuitionistic fuzzy sets by Atanassov [5]. The concept of intuitionistic fuzzy set was introduced and studied by Atanassov [5, 6] as a generalization of the notion of fuzzy set. Some applications of intuitionistic fuzzy sets are discussed in [7]. In [8–12], Davvaz et al. applied the concept of intuitionistic fuzzy sets to algebraic hyperstructures and some related properties are investigated.

The purpose of this paper is to deal with the algebraic structure of ring by applying intuitionistic fuzzy soft theory. We introduce the notion of intuitionistic fuzzy soft ring and study some of its characterization of operations and algebraic properties.

2. Preliminaries

In this section, for the sake of completeness, we first cite some useful definitions and results.

Definition 1 (see [13]). A fuzzy subset 𝜇 in a set 𝑋 is a function 𝜇∶𝑋→[0,1].

Definition 2 (see [1]). Let 𝑈 be an initial universe and 𝐸 a set of parameters. Let 𝑃(𝑈) denote the power set of 𝑈 and 𝐴⊂𝐸. A pair (𝐹,𝐴) is called a soft set over 𝑈, where 𝐹 is a mapping given by 𝐹∶𝐴→𝑃(𝑈). In other words, a soft set over 𝑈 is a parameterized family of subsets of the universe 𝑈.

Definition 3 (see [14]). Let 𝑈 be an initial universe, and 𝐸 a set of parameters and 𝐹𝑆(𝑈) denotes the fuzzy power set of 𝑈 and 𝐴⊂𝐸. A pair (𝐹,𝐴) is called a fuzzy soft set over 𝑈, where 𝐹 is a mapping given by 𝐹∶𝐴→𝐹𝑆(𝑈). A fuzzy soft set is a parameterized family of fuzzy subsets of 𝑈.

Definition 4 (see [14]). An intuitionistic fuzzy set 𝐴 on the universe 𝑋 can be defined as follows: 𝐴=𝑥,𝜇_𝐴(𝑥),𝑣_𝐴(𝑥)∣𝑥∈𝑋, (1) where 𝜇_𝐴(𝑥)∶𝑋→[0,1] and 𝑣_𝐴(𝑥)∶𝑋→[0,1] with the following property: 0≤𝜇_𝐴(𝑥)+𝑣_𝐴(𝑥)≤1,∀𝑥∈𝑋. (2) The values 𝜇_𝐴(𝑥) and 𝑣_𝐴(𝑥) denote the degree of membership and nonmembership of 𝑥 to 𝐴, respectively.

Lemma 5 (see [7, 15]). Let 𝐴={⟨𝑥,𝜇_𝐴(𝑥),𝑣_𝐴(𝑥)⟩∣𝑥∈𝑋} be an intuitionistic fuzzy set in 𝑋 and let 𝑟,𝑡∈[0,1] such that 𝑟+𝑡≤1. One defines (1)□𝐴={⟨𝑥,𝜇_𝐴(𝑥),𝜇^𝑐_𝐴(𝑥)⟩∣𝑥∈𝑋}, (2)⋄𝐴={⟨𝑥,𝑣^𝑐_𝐴(𝑥),𝑣_𝐴(𝑥)⟩∣𝑥∈𝑋}, (3)𝑃_𝑟,𝑡(𝐴)={(𝑥,max{𝑟,𝜇_𝐴(𝑥)},min{𝑡,𝑣_𝐴(𝑥)})∣𝑥∈𝑋}, (4)𝑄_𝑟,𝑡(𝐴)={(𝑥,min{𝑟,𝜇_𝐴(𝑥)},max{𝑡,𝑣_𝐴(𝑥)})∣𝑥∈𝑋}. Then, □𝐴, ⋄𝐴, 𝑃_𝑟,𝑡(𝐴), and 𝑄_𝑟,𝑡(𝐴) are intuitionistic fuzzy sets in 𝑋.

Definition 6 (see [16]). Let 𝑈 be an initial universe and 𝐸 a set of parameters and IFS(𝑈) denotes the intuitionistic fuzzy power set of 𝑈 and 𝐴⊂𝐸. A pair (𝐹,𝐴) is called an intuitionistic fuzzy soft set over 𝑈, where 𝐹 is a mapping given by 𝐹∶𝐴→IFS(𝑈).
An intuitionistic fuzzy soft set is a parameterized family of intuitionistic fuzzy subsets of 𝑈; a fuzzy soft set is a special case of an intuitionistic fuzzy soft set, because when all the intuitionistic fuzzy subset of 𝑈 degenerates into fuzzy subsets, the corresponding intuitionistic fuzzy soft set degenerates into a fuzzy soft set.

In general, for all 𝜀∈𝐴⊆𝐸, 𝐹(𝜀) is an intuitionistic fuzzy set on 𝑈, which is called the intuitionistic fuzzy set of parameter 𝜀. The intuitionistic value 𝜇_𝐹(𝜀)(𝑥),𝑣_𝐹(𝜀)(𝑥) (3) denotes the degree that object 𝑥∈𝑈 holds parameter 𝜀. 𝐹(𝜀) can be written as following: 𝐹(𝜀)=𝑥,𝜇_𝐹(𝜀)(𝑥),𝑣_𝐹(𝜀)(𝑥)∣𝑥∈𝑈. (4)

If for all 𝑥∈𝑈, 𝜇_𝐹(𝜀)(𝑥)+𝑣_𝐹(𝜀)(𝑥)=1, then 𝐹(𝜀) degenerates into a fuzzy set; if for all 𝑥∈𝑈 and for all 𝜀∈𝐴⊆𝐸, 𝜇_𝐹(𝜀)(𝑥)+𝑣_𝐹(𝜀)(𝑥)=1, then the intuitionistic fuzzy soft set (𝐹,𝐴) degenerates into a fuzzy soft set.

Example 7. Let (𝐹,𝐴) describe the character of the students with respect to the given parameter, for finding the best student of an academic year. Let the set of students under consideration be 𝑈={𝑠₁,𝑠₂,𝑠₃,𝑠₄}. Let 𝐴⊆𝐸 and .𝐴=𝑟=“result”,𝑐=“conduct”,𝑔=“gamesandsportsperformances” (5) Let 𝐹(𝑟)=𝑠₁,0.8,0.1,𝑠₂,𝑠,0.9,0.05₃,0.85,0.1,𝑠₄,𝐹(𝑐)=𝑠,0.75,0.2₁,0.6,0.3,𝑠₂,𝑠,0.65,0.2₃,0.7,0.2,𝑠₄,𝐹𝑔=𝑠,0.65,0.2₁,0.75,0.2,𝑠₂,𝑠,0.5,0.3₃,0.5,0.4,𝑠₄,0.7,0.2. (6)
Then, the family {𝐹(𝑟),𝐹(𝑐),𝐹(𝑔)} of IFS(𝑈) is an intuitionistic fuzzy soft set.

Definition 8 (see [16]). Let (𝐹,𝐴) and (𝐺,𝐵) be two intuitionistic fuzzy soft sets over 𝑈. Then, (𝐹,𝐴) is said to be an intuitionistic fuzzy soft subset of (𝐺,𝐵) if(1)𝐴⊂𝐵,(2)𝐹(𝜀) is an intuitionistic fuzzy subset of 𝐺(𝜀), for all 𝜀∈𝐴.
We denote the above inclusion relationship by (𝐹,𝐴)⊆(𝐺,𝐵). Similarly, (𝐹,𝐴) is called an intuitionistic fuzzy soft superset of (𝐺,𝐵) if (𝐺,𝐵) is an intuitionistic fuzzy soft subset of (𝐹,𝐴). We denoted the above relationship by (𝐹,𝐴)⊃(𝐺,𝐵). (𝐹,𝐴) and (𝐺,𝐵) over a universe 𝑈 are said to be intuitionistic fuzzy soft equals if (𝐹,𝐴)⊆(𝐺,𝐵) and (𝐺,𝐵)⊆(𝐹,𝐴).

Definition 9 (see [16]). Let (𝐹,𝐴) and (𝐺,𝐵) be two intuitionistic fuzzy soft sets over a universe 𝑈. Then, “(𝐹,𝐴) and (𝐺,𝐵)” is denoted by (𝐹,𝐴)∧(𝐺,𝐵) and is defined by (𝐹,𝐴)∧(𝐺,𝐵)=(𝐻,𝐴×𝐵), where 𝐻(𝛼,𝛽)=𝐹(𝛼)∩𝐺(𝛽), for all (𝛼,𝛽)∈𝐴×𝐵.

Definition 10 (see [16]). Let (𝐹,𝐴) and (𝐺,𝐵) be two intuitionistic fuzzy soft sets over a universe 𝑈. Then, “(𝐹,𝐴) or (𝐺,𝐵)” is denoted by (𝐹,𝐴)∨(𝐺,𝐵) and is defined by (𝐹,𝐴)∨(𝐺,𝐵)=(𝐻,𝐴×𝐵), where 𝐻(𝛼,𝛽)=𝐹(𝛼)∪𝐺(𝛽), for all (𝛼,𝛽)∈𝐴×𝐵.

Definition 11 (see [16]). The intersection of two intuitionistic fuzzy soft sets (𝐹,𝐴) and (𝐺,𝐵) over a universe 𝑈 is an intuitionistic fuzzy soft set denoted by (𝐻,𝐶), where 𝐶=𝐴∪𝐵 and 𝐻(𝜀)=𝐹(𝜀)𝐺(𝜀)𝐹(𝜀),𝐺(𝜀if𝜀∈𝐴−𝐵,if𝜀∈𝐵−𝐴,min)if𝜀∈𝐴∩𝐵, (7) for all 𝜀∈𝐶. This is denoted by (𝐻,𝐶)=(𝐹,𝐴)∩(𝐺,𝐵).

Definition 12 (see [16]). The union of two intuitionistic fuzzy soft sets (𝐹,𝐴) and (𝐺,𝐵) over a universe 𝑈 is an intuitionistic fuzzy soft set denoted by (𝐻,𝐶), where 𝐶=𝐴∪𝐵 and 𝐻(𝜀)=𝐹(𝜀)𝐺(𝜀)𝐹(𝜀),𝐺(𝜀if𝜀∈𝐴−𝐵,if𝜀∈𝐵−𝐴,max)if𝜀∈𝐴∩𝐵, (8) for all 𝜀∈𝐶. This is denoted by (𝐻,𝐶)=(𝐹,𝐴)∪(𝐺,𝐵).
In contrast with the above definitions of union and intersection of intuitionistic fuzzy soft sets, we may sometimes adopt different definitions of union and intersection given as follows.

Definition 13 (see [16]). Let (𝐹,𝐴) and (𝐺,𝐵) be two intuitionistic fuzzy soft sets over a universe 𝑈 such that 𝐴∩𝐵≠∅. The biunion of (𝐹,𝐴) and (𝐺,𝐵) is defined to be the intuitionistic fuzzy soft set (𝐻,𝐶), where 𝐶=𝐴∩𝐵 and 𝐻(𝜀)=𝐹(𝜀)∪𝐺(𝜀) for all 𝜀∈𝐶. This is denoted by (𝐻,𝐶)=(𝐹,𝐴)⊔(𝐺,𝐵).

Definition 14 (see [16]). Let (𝐹,𝐴) and (𝐺,𝐵) be two intuitionistic fuzzy soft sets over a universe 𝑈 such that 𝐴∩𝐵≠∅. The bi-intersection (𝐹,𝐴) and (𝐺,𝐵) is defined to be the intuitionistic fuzzy soft set (𝐻,𝐶), where 𝐶=𝐴∩𝐵 and 𝐻(𝜀)=𝐹(𝜀)∩𝐺(𝜀) for all 𝜀∈𝐶. This is denoted by (𝐻,𝐶)=(𝐹,𝐴)⊓(𝐺,𝐵).

Definition 15. A fuzzy set 𝜇 in ring 𝑅 is called a fuzzy ideal of 𝑅, if for all 𝑥,𝑦∈𝑅 the following conditions hold:(1)𝜇(𝑥−𝑦)≥min{𝜇(𝑥),𝜇(𝑦)}, (2)𝜇(𝑥𝑦)≥max{𝜇(𝑥),𝜇(𝑦)}.

3. Intuitionistic Fuzzy Soft Rings

Definition 16. A pair (𝐹,𝐴) is called an intuitionistic fuzzy soft ring over 𝑅, where 𝐹 is a mapping given by 𝐹∶𝐴⟶([0,1]×[0,1])^𝑅,𝐹(𝜀)[]×[],𝐹(𝜀)=∶𝑅⟶0,10,1𝑥,𝜇_𝐹(𝜀)(𝑥),𝑣_𝐹(𝜀)(𝑥)∶𝑥∈𝑅∀𝜀∈𝐴, (9) if for all 𝑥,𝑦∈𝑅 the following conditions hold: (1)𝜇_𝐹(𝜀)(𝑥−𝑦)≥𝜇_𝐹(𝜀)(𝑥)∧𝜇_𝐹(𝜀)(𝑦) and 𝑣_𝐹(𝜀)(𝑥−𝑦)≤𝑣_𝐹(𝜀)(𝑥)∨𝑣_𝐹(𝜀)(𝑦),(2)𝜇_𝐹(𝜀)(𝑥𝑦)≥𝜇_𝐹(𝜀)(𝑥)∧𝜇_𝐹(𝜀)(𝑦) and 𝑣_𝐹(𝜀)(𝑥𝑦)≤𝑣_𝐹(𝜀)(𝑥)∨𝑣_𝐹(𝜀)(𝑦).

Definition 17. For two intuitionistic fuzzy soft rings (𝐹,𝐴) and (𝐺,𝐵) over a universe 𝑅, we say that (𝐹,𝐴) is an intuitionistic fuzzy soft subring of (𝐺,𝐵) and write (𝐹,𝐴)⊆(𝐺,𝐵) if(i)𝐴⊆𝐵, (ii)for each 𝑥∈𝑅, 𝜀∈𝐴, 𝜇_𝐹(𝜀)(𝑥)≤𝜇_𝐺(𝜀)(𝑥) and 𝑣_𝐹(𝜀)(𝑥)≥𝑣_𝐺(𝜀)(𝑥).

Definition 18. Two intuitionistic fuzzy soft rings (𝐹,𝐴) and (𝐺,𝐵) over a universe 𝑅 are said to be equal if (𝐹,𝐴)⊆(𝐺,𝐵) and (𝐺,𝐵)⊆(𝐹,𝐴).

Theorem 19. Let (𝐹,𝐴) and (𝐺,𝐵) be two intuitionistic fuzzy soft rings over a universe 𝑅. If 𝐹(𝜀)≤𝐺(𝜀) for all 𝜀∈𝐴, then (𝐹,𝐴) is an intuitionistic fuzzy soft subring of (𝐺,𝐵).

Proof. The proof is straightforward.

Definition 20. The union of two intuitionistic fuzzy soft rings (𝐹,𝐴) and (𝐺,𝐵) over a universe 𝑅 is denoted by (𝐹,𝐴)∪(𝐺,𝐵)and is defined by an intuitionistic fuzzy soft ring 𝐻∶𝐴∪𝐵→[0,1]^𝑅 such that for each 𝜀∈𝐴∪𝐵, 𝐻(𝜀)=𝑥,𝜇_𝐹(𝜀)(𝑥),𝑣_𝐹(𝜀)(𝑥)if𝜀∈𝐴−𝐵𝑥,𝜇_𝐺(𝜀)(𝑥),𝑣_𝐺(𝜀)(𝑥)if𝜀∈𝐵−𝐴𝑥,𝜇_𝐹(𝜀)(𝑥)∨𝜇_𝐺(𝜀)(𝑥),𝑣_𝐹(𝜀)(𝑥)∧𝑣_𝐺(𝜀)(𝑥)if𝜀∈𝐴∩𝐵. (10)
This is denoted by (𝐻,𝐶)=(𝐹,𝐴)∪(𝐺,𝐵), where 𝐶=𝐴∪𝐵.

Theorem 21. If (𝐹,𝐴) and (𝐺,𝐵) are two intuitionistic fuzzy soft rings over a universe 𝑅, then, so are (𝐹,𝐴)∪(𝐺,𝐵).

Proof. For any 𝜀=(𝛾,𝛽)∈𝐴∪𝐵 and 𝑥,𝑦∈𝑅, we consider the following cases.
Case 1. Let 𝜀∈𝐴−𝐵. Then, 𝜇_𝐻(𝜀)𝑥−𝑦=𝜇_𝐹(𝜀)𝑥−𝑦≥𝜇_𝐹(𝜀)(𝑥)∧𝜇_𝐹(𝜀)𝑦=𝜇_𝐻(𝜀)(𝑥)∧𝜇_𝐻(𝜀)𝑦,𝜇_𝐻(𝜀)𝑥𝑦=𝜇_𝐹(𝜀)𝑥𝑦≥𝜇_𝐹(𝜀)(𝑥)∧𝜇_𝐹(𝜀)𝑦=𝜇_𝐻(𝜀)(𝑥)∧𝜇_𝐻(𝜀)𝑦,𝑣_𝐻(𝜀)𝑥−𝑦=𝑣_𝐹(𝜀)𝑥−𝑦≤𝑣_𝐹(𝜀)(𝑥)∨𝑣_𝐹(𝜀)𝑦=𝑣_𝐻(𝜀)(𝑥)∨𝑣_𝐻(𝜀)𝑦,𝑣_𝐻(𝜀)𝑥𝑦=𝑣_𝐹(𝜀)𝑥𝑦≤𝑣_𝐹(𝜀)(𝑥)∨𝑣_𝐹(𝜀)𝑦=𝑣_𝐻(𝜀)(𝑥)∨𝑣_𝐻(𝜀)𝑦. (11)
Case 2. Let 𝜀∈𝐵−𝐴. Then, analogous to the proof of Case  1, we have 𝜇_𝐻(𝜀)𝑥−𝑦≥𝜇_𝐻(𝜀)(𝑥)∧𝜇_𝐻(𝜀)𝑦,𝜇_𝐻(𝜀)𝑥𝛼𝑦≥𝜇_𝐻(𝜀)(𝑥)∧𝜇_𝐻(𝜀)𝑦,𝑣_𝐻(𝜀)𝑥−𝑦≤𝑣_𝐻(𝜀)(𝑥)∨𝑣_𝐻(𝜀)𝑦,𝑣_𝐻(𝜀)𝑥𝑦≤𝑣_𝐻(𝜀)(𝑥)∨𝑣_𝐻(𝜀)𝑦. (12)

Case 3. Let 𝜀∈𝐴∩𝐵. In this case the proof is straightforward.

Thus, in any cases, we have 𝜇_𝐻(𝜀)𝑥−𝑦≥𝜇_𝐻(𝜀)(𝑥)∧𝜇_𝐻(𝜀)𝑦,𝜇_𝐻(𝜀)𝑥𝑦≥𝜇_𝐻(𝜀)(𝑥)∧𝜇_𝐻(𝜀)𝑦,𝑣_𝐻(𝜀)𝑥−𝑦≤𝑣_𝐻(𝜀)(𝑥)∨𝑣_𝐻(𝜀)𝑦,𝑣_𝐻(𝜀)𝑥𝑦≤𝑣_𝐻(𝜀)(𝑥)∨𝑣_𝐻(𝜀)𝑦. (13)

Therefore, (𝐹,𝐴)∪(𝐺,𝐵) is an intuitionistic fuzzy soft ring.

Definition 22. The intersection of two intuitionistic fuzzy soft rings (𝐹,𝐴) and (𝐺,𝐵) over a universe 𝑅 is denoted by (𝐹,𝐴)∩(𝐺,𝐵) and is defined by 𝐻(𝜀)=𝑥,𝜇_𝐹(𝜀)(𝑥),𝑣_𝐹(𝜀)(𝑥)if𝜀∈𝐴−𝐵𝑥,𝜇_𝐺(𝜀)(𝑥),𝑣_𝐺(𝜀)(𝑥𝜇)if𝜀∈𝐵−𝐴_𝐹(𝜀)(𝑥)∧𝜇_𝐺(𝜀)(𝑥),𝑣_𝐹(𝜀)(𝑥)∨𝑣_𝐺(𝜀)(𝑥)if𝜀∈𝐴∩𝐵. (14)
This is denoted by (𝐻,𝐶)=(𝐹,𝐴)∩(𝐺,𝐵), where 𝐶=𝐴∪𝐵.

Theorem 23. If (𝐹,𝐴) and (𝐺,𝐵) are two intuitionistic fuzzy soft rings over a universe 𝑅, then, so are (𝐹,𝐴)∩(𝐺,𝐵).

Proof. The proof is straightforward.

Definition 24. Let (𝐹,𝐴) and (𝐺,𝐵) be two intuitionistic fuzzy soft rings over a universe 𝑅. Then, “(𝐹,𝐴) AND (𝐺,𝐵)" is denoted by (𝐹,𝐴)∧(𝐺,𝐵) and is defined by (𝐹,𝐴)∧(𝐺,𝐵)=(𝐻,𝐶), where 𝐶=𝐴×𝐵 and 𝐻∶𝐶→([0,1]×[0,1])^𝑅is defined as 𝐻𝛼,𝛽=𝐹(𝛼)∩𝐺𝛽,∀𝛼,𝛽∈𝐶. (15)

Theorem 25. If (𝐹,𝐴) and (𝐺,𝐵) are two intuitionistic fuzzy soft rings over a universe 𝑅, then, so are (𝐹,𝐴)∧(𝐺,𝐵) and (𝐹,𝐴)⊓(𝐺,𝐵).

Proof. For all 𝑥,𝑦∈𝑅 and (𝛼,𝛽)∈𝐴×𝐵 we have 𝜇_{𝐻(𝛼,𝛽)}𝑥−𝑦=𝜇_𝐹(𝛼)𝑥−𝑦∩𝜇_𝐺(𝛽)≥𝜇𝑥−𝑦_𝐹(𝛼)(𝑥)∧𝜇_𝐹(𝛼)𝑦∩𝜇_𝐺(𝛽)(𝑥)∧𝜇_𝐺(𝛽)𝑦=𝜇_𝐹(𝛼)(𝑥)∩𝜇_𝐺(𝛽)(𝑥)∧𝜇_𝐹(𝛼)𝑦∩𝜇_𝐺(𝛽)𝑦=𝜇_{𝐻(𝛼,𝛽)}(𝑥)∧𝜇_{𝐻(𝛼,𝛽)}𝑦,𝜇_{𝐻(𝛼,𝛽)}𝑥𝑦=𝜇_𝐹(𝛼)𝑥𝑦∩𝜇_𝐺(𝛽)≥𝜇𝑥𝑦_𝐹(𝛼)(𝑥)∧𝜇_𝐹(𝛼)𝑦∩𝜇_𝐺(𝛽)(𝑥)∧𝜇_𝐺(𝛽)𝑦=𝜇_𝐹(𝛼)(𝑥)∩𝜇_𝐺(𝛽)(𝑥)∧𝜇_𝐹(𝛼)𝑦∩𝜇_𝐺(𝛽)𝑦=𝜇_{𝐻(𝛼,𝛽)}(𝑥)∧𝜇_{𝐻(𝛼,𝛽)}𝑦. (16) In a similar way, we have 𝑣_{𝐻(𝛼,𝛽)}𝑥−𝑦≤𝑣_{𝐻(𝛼,𝛽)}(𝑥)∨𝑣_{𝐻(𝛼,𝛽)}𝑦,𝑣_{𝐻(𝛼,𝛽)}𝑥𝑦≤𝑣_{𝐻(𝛼,𝛽)}(𝑥)∨𝑣_{𝐻(𝛼,𝛽)}𝑦. (17)
The case for (𝐹,𝐴)⊓(𝐺,𝐵) can be similarly proved.

Example 26 (see [17]). Consider the ring 𝑅={ℎ₁,ℎ₂,ℎ₃,ℎ₄} with the following tables: +ℎ₁ℎ₂ℎ₃ℎ₄ℎ₁ℎ₁ℎ₂ℎ₃ℎ₄ℎ₂ℎ₂ℎ₃ℎ₄ℎ₁ℎ₃ℎ₃ℎ₄ℎ₁ℎ₂ℎ₄ℎ₄ℎ₁ℎ₂ℎ₃•ℎ₁ℎ₂ℎ₃ℎ₄ℎ₁ℎ₁ℎ₁ℎ₁ℎ₁ℎ₂ℎ₁ℎ₂ℎ₃ℎ₄ℎ₃ℎ₁ℎ₃ℎ₁ℎ₃ℎ₄ℎ₁ℎ₄ℎ₃ℎ₂ (18)
Let 𝐴={𝑒₁,𝑒₂,𝑒₃} and 𝐹∶𝐴→ℱ(𝑅) be a set-valued function defined by 𝐹𝑒₁=ℎ₁,0.84,0.11,ℎ₂,ℎ,0.56,0.31₃,0.83,0.12,ℎ₄,𝐹𝑒,0.69,0.15₂=ℎ₁,0.76,0.21,ℎ₂,ℎ,0.62,0.24₃,0.90,0.05,ℎ₄,𝐹𝑒,0.75,0.20₃=ℎ₁,0.94,0.04,ℎ₂,ℎ,0.74,0.17₃,0.68,0.26,ℎ₄,0.86,0.13. (19)
Obviously, (𝐹,𝐴) is an intuitionistic fuzzy soft set over 𝑅. Moreover, we see that ⟨𝜇_𝐹(𝜀)(𝑥),𝑣_𝐹(𝜀)(𝑥)⟩ is an intuitionistic fuzzy subring of 𝑅, for all 𝑥∈𝐴. Therefore, (𝐹,𝐴) is an intuitionistic fuzzy soft ring over 𝑅.

Definition 27. Let (𝐹,𝐴) and (𝐺,𝐵) be two intuitionistic fuzzy soft rings over a universe 𝑅. Then, “(𝐹,𝐴) OR (𝐺,𝐵)” is denoted by (𝐹,𝐴)∨(𝐺,𝐵) and is defined by (𝐹,𝐴)∨(𝐺,𝐵)=(𝐻,𝐶), where 𝐶=𝐴×𝐵 and 𝐻∶𝐶→([0,1]×[0,1])^𝑅 is defined as 𝐻𝛼,𝛽=𝐹(𝛼)∪𝐺𝛽,∀𝛼,𝛽∈𝐶. (20)

Theorem 28. If (𝐹,𝐴) and (𝐺,𝐵) are two intuitionistic fuzzy soft rings over a universe 𝑅, then, so are (𝐹,𝐴)∨(𝐺,𝐵) and (𝐹,𝐴)⊔(𝐺,𝐵).

Proof. The proof is straightforward.

The following theorem is a generalization of previous results.

Theorem 29. Let (𝐹,𝐴) be an intuitionistic fuzzy soft rings over a universe 𝑅, and let {(𝐹_𝑖,𝐴_𝑖)}_𝑖∈𝐼 be a nonempty family of intuitionistic fuzzy soft rings. Then, one has the following.(1)Π_𝑖∈𝐼(𝐹_𝑖,𝐴_𝑖) is an intuitionistic fuzzy soft ring over 𝑅.(2)∧_𝑖∈𝐼(𝐹_𝑖,𝐴_𝑖) is an intuitionistic fuzzy soft ring over 𝑅.(3)If 𝐴_𝑖∩𝐴_𝑗=∅, for all 𝑖,𝑗∈𝐼, then ∨_𝑖∈𝐼(𝐹_𝑖,𝐴_𝑖) is an intuitionistic fuzzy soft ring over 𝑅.

Theorem 30. Let 𝑁=(𝐹,𝐴) be an intuitionistic fuzzy soft ring over a universe 𝑅, and let 𝑟,𝑡∈[0,1] such that 𝑟+𝑡≤1. Then the following are intuitionistic fuzzy soft rings: (□𝐹,𝐴),(⋄𝐹,𝐴),(𝑃_𝑟,𝑡(𝐹),𝐴),(𝑄_𝑟,𝑡(𝐹),𝐴) over 𝑅, where □𝐹,𝐴=□𝐹,𝐴=□𝑁=𝜇_𝐹(𝜀)(𝑥),𝜇^𝑐_𝐹(𝜀)(𝑥),⋄𝑣,𝐴𝐹,𝐴=⋄𝐹,𝐴=⋄𝑁=_𝐹(𝜀)(𝑥),𝑣^𝑐_𝐹(𝜀)(𝑥),𝑃,𝐴_𝑟,𝑡𝐹,𝐴=𝑃_𝑟,𝑡𝐹,𝐴,𝑄_𝑟,𝑡𝐹,𝐴=𝑄_𝑟,𝑡𝐹,𝐴. (21)

Proof. Suppose that (𝐹,𝐴) is an intuitionistic fuzzy soft ring over 𝑅. Then, □𝐹,𝐴=𝜇_𝐹(𝜀)(𝑥),𝜇^𝑐_𝐹(𝜀)(𝑥),𝐴. (22)
For all 𝑥,𝑦∈𝑅, 𝜇_𝐹(𝜀)𝑥−𝑦≥𝜇_𝐹(𝜀)(𝑥)∧𝜇_𝐹(𝜀)𝑦,𝜇_𝐹(𝜀)𝑥𝑦≥𝜇_𝐹(𝜀)(𝑥)∧𝜇_𝐹(𝜀)𝑦,𝜇^𝑐_𝐹(𝜀)𝑥−𝑦=1−𝜇_𝐹(𝜀)≤𝑥−𝑦1−𝜇_𝐹(𝜀)(𝑥)∧1−𝜇_𝐹(𝜀)𝑦=𝜇^𝑐_𝐹(𝜀)(𝑥)∧𝜇^𝑐_𝐹(𝜀)𝑦,𝜇^𝑐_𝐹(𝜀)𝑥𝑦=1−𝜇_𝐹(𝜀)≤𝑥𝑦1−𝜇_𝐹(𝜀)(𝑥)∧1−𝜇_𝐹(𝜀)𝑦=𝜇^𝑐_𝐹(𝜀)(𝑥)∧𝜇^𝑐_𝐹(𝜀)𝑦. (23)
The proof of other cases is similar.

Definition 31. Let (𝐹,𝐴) and (𝐺,𝐵) be two intuitionistic fuzzy soft rings over a universe 𝑅. Then, the product of (𝐹,𝐴) and (𝐺,𝐵) is defined to be the intuitionistic fuzzy soft ring (𝐹∘𝐺,𝐶), where 𝐶=𝐴∪𝐵 and 𝜇_𝐹∘𝐺_(𝜀)(𝑥)=𝜇_𝐹(𝜀)(𝑥)𝜇if𝜀∈𝐴−𝐵_𝐺(𝜀)(𝑥)⋁if𝜀∈𝐵−𝐴_{𝑥=𝑎𝑏}𝜇_𝐹(𝜀)(𝑎)∧𝜇_𝐺(𝜀)(𝑏𝑣)if𝜀∈𝐴∩𝐵,_𝐹∘𝐺_(𝜀)(𝑥)=𝑣_𝐹(𝜀)(𝑥)𝑣if𝜀∈𝐴−𝐵_𝐺(𝜀)(𝑥)⋀if𝜀∈𝐵−𝐴_{𝑥=𝑎𝑏}𝑣_𝐹(𝜀)(𝑎)∨𝑣_𝐺(𝜀)(𝑏)if𝜀∈𝐴∩𝐵, (24)(25) for all 𝜀∈𝐶 and 𝑥∈𝑅. This is denoted by (𝐹∘𝐺,𝐶)=(𝐹,𝐴)∘(𝐺,𝐵).

Theorem 32. If (𝐹,𝐴) and (𝐺,𝐵) are intuitionistic fuzzy soft rings over a universe 𝑅, then so is (𝐹,𝐴)∘(𝐺,𝐵).

Proof. Let (𝐹,𝐴) and (𝐺,𝐵) be two intuitionistic fuzzy soft rings over 𝑅. Then, for any 𝜀∈𝐴∪𝐵 and 𝑥,𝑦∈𝑅, we consider the following cases.
Case  1. Let 𝜀∈𝐴−𝐵. Then, 𝜇_𝐹∘𝐺^(𝜀)𝑥−𝑦=𝜇_𝐹(𝜀)𝑥−𝑦≥𝜇_𝐹(𝜀)(𝑥)∧𝜇_𝐹(𝜀)𝑦=𝜇_𝐹∘𝐺_(𝜀)(𝑥)∧𝜇_𝐹∘𝐺_(𝜀)𝑦,𝜇_𝐹∘𝐺_(𝜀)𝑥𝑦=𝜇_𝐹(𝜀)𝑥𝑦≥𝜇_𝐹(𝜀)(𝑥)∧𝜇_𝐹(𝜀)𝑦=𝜇_𝐹∘𝐺_(𝜀)(𝑥)∧𝜇_𝐹∘𝐺_(𝜀)𝑦,𝑣_𝐹∘𝐺^(𝜀)𝑥−𝑦=𝑣_𝐹(𝜀)𝑥−𝑦≤𝑣_𝐹(𝜀)(𝑥)∨𝑣_𝐹(𝜀)𝑦=𝑣_𝐹∘𝐺^(𝜀)(𝑥)∨𝑣_𝐹∘𝐺^(𝜀)𝑦,𝑣_𝐹∘𝐺^(𝜀)𝑥𝑦=𝑣_𝐹(𝜀)𝑥𝑦≤𝑣_𝐹(𝜀)(𝑥)∨𝑣_𝐹(𝜀)𝑦=𝑣_𝐹∘𝐺_(𝜀)(𝑥)∨𝑣_𝐹∘𝐺_(𝜀)𝑦. (26)
Case  2. Let 𝜀∈𝐵−𝐴. Then, analogous to the proof of Case  1, the proof is straightforward.
Case  3. Let 𝜀∈𝐴∩𝐵. Then, 𝜇_𝐹∘𝐺_(𝜀)(𝑥)=_{𝑥=𝑎𝑏}𝜇_𝐹(𝜀)(𝑎)∧𝜇_𝐺(𝜀)(𝑏)≤_{𝑥𝑦=𝑎𝑏}𝜇_𝐹(𝜀)(𝑎)∧𝜇_𝐺(𝜀)≤𝑏𝑦_{𝑥𝑦=𝑐𝑑}𝜇_𝐹(𝜀)(𝑐)∧𝜇_𝐺(𝜀)(𝑑)=𝜇_𝐹∘𝐺_(𝜀)𝑥𝑦. (27)
Similarly, we have 𝜇_𝐹∘𝐺_(𝜀)(𝑥𝑦)≥𝜇_𝐹∘𝐺_(𝜀)(𝑦), and so 𝜇_𝐹∘𝐺^(𝜀)𝑥𝑦≥𝜇_𝐹∘𝐺^(𝜀)(𝑥)∧𝜇_𝐹∘𝐺^(𝜀)𝑦. (28)
In a similar way, we prove that 𝑣_𝐹∘𝐺^(𝜀)𝑥𝑦≤𝑣_𝐹∘𝐺^(𝜀)(𝑥)∨𝑣_𝐹∘𝐺^(𝜀)𝑦, (29) and the rest of the other cases can be similarly proved. Therefore, (𝐹,𝐴)∘(𝐺,𝐵) is an intuitionistic fuzzy soft ring over 𝑅.

Definition 33. Let 𝐼 be an ideal of 𝑅 and let (𝐹,𝐼) be an intuitionistic fuzzy soft set over 𝑅. Then, (𝐹,𝐼) is an intuitionistic fuzzy soft ideal of 𝑅, which will be denoted by (𝐹,𝐼)⊳𝑅, if for each 𝑥,𝑦∈𝐼 and 𝑟∈𝑅 the following conditions hold:(i)𝜇_𝐹(𝜀)(𝑥−𝑦)≥𝜇_𝐹(𝜀)(𝑥)∧𝜇_𝐹(𝜀)(𝑦) and 𝑣_𝐹(𝜀)(𝑥−𝑦)≤𝑣_𝐹(𝜀)(𝑥)∨𝑣_𝐹(𝜀)(𝑦).

Theorem 34. One has(ii)𝜇_𝐹(𝜀)(𝑥𝑟)≥𝜇_𝐹(𝜀)(𝑥) and 𝑣_𝐹(𝜀)(𝑥𝑟)≤𝑣_𝐹(𝜀)(𝑥),(iii)𝜇_𝐹(𝜀)(𝑟𝑥)≥𝜇_𝐹(𝜀)(𝑥) and 𝑣_𝐹(𝜀)(𝑟𝑥)≤𝑣_𝐹(𝜀)(𝑥).

Theorem 35. For any intuitionistic fuzzy soft ideals (𝐹,𝐼) and (𝐺,𝐽) over 𝑅, where 𝐼∩𝐽≠∅, one has (𝐹,𝐼)∩(𝐺,𝐽)⊳𝑅.

Proof. Since 𝐼,𝐽⊳𝑅, then 𝐼∩𝐽⊳𝑅. By Definition  2.5, we can write (𝐹,𝐼)∩(𝐺,𝐽)=(𝐻,𝑆), where 𝑆=𝐼∩𝐽, and for all 𝜀∈𝑆=𝐼∩𝐽,𝜇_𝐻(𝜀)=𝜇_𝐹(𝜀)∩𝜇_𝐺(𝜀). Then, for all 𝑥,𝑦∈𝐼∩𝐽 and 𝑟∈𝑅, 𝜇_𝐻(𝜀)𝑥−𝑦=𝜇_𝐹(𝜀)𝑥−𝑦∩𝜇_𝐺(𝜀)≥𝜇𝑥−𝑦_𝐹(𝜀)(𝑥)∧𝜇_𝐹(𝜀)𝑦∩𝜇_𝐺(𝜀)(𝑥)∧𝜇_𝐺(𝜀)𝑦=𝜇_𝐹(𝜀)(𝑥)∧𝜇_𝐺(𝜀)(𝑥)∧𝜇_𝐹(𝜀)𝑦∧𝜇_𝐺(𝜀)𝑦. (30)
Since (𝐹,𝐼)⊳𝑅 and (𝐺,𝐽)⊳𝑅, we know that 𝜇_𝐹(𝜀) and 𝜇_𝐺(𝜀) are intuitionistic fuzzy soft ideals of 𝑅. For all 𝜀∈𝑆, we have 𝜇_𝐻(𝜀)𝑥−𝑦≥𝜇_𝐻(𝜀)(𝑥)∧𝜇_𝐻(𝜀)𝑦,𝜇_𝐻(𝜀)(𝑥𝑟)=𝜇_𝐹(𝜀)(𝑥𝑟)∩𝜇_𝐺(𝜀)(𝑥𝑟)≥𝜇_𝐹(𝜀)(𝑥)∩𝜇_𝐺(𝜀)(𝑥)=𝜇_𝐻(𝜀)(𝑥). (31)
We obtain a similar result for 𝑣_𝐹(𝜀). Hence, (𝐻,𝑆) is an intuitionistic fuzzy soft ideal. Therefore, (𝐹,𝐼)∩(𝐺,𝐽)=(𝐻,𝑆)⊳𝑅.

Theorem 36. For any fuzzy soft ideals (𝑓,𝐼) and (𝑔,𝐽) over 𝑅, in which 𝐼 and 𝐽 are disjoint, one has (𝑓,𝐼)∪(𝑔,𝐽)⊳𝑅.

Proof. Assume that (𝑓,𝐼)⊳𝑅 and (𝑔,𝐽)⊳𝑅. By means of definition we can write (𝑓,𝐼)∪(𝑔,𝐽)=(ℎ,𝐾), where 𝐾=𝐼∪𝐽 and for every 𝜇∈𝐾ℎ_𝜇(𝑐)=𝑓_𝜇𝑔if𝑐∈𝐼−𝐽_𝜇𝑓if𝑐∈𝐽−𝐼_𝜇∨𝑔_𝜇if𝑐∈𝐼∩𝐽. (32)
Since 𝐼∩𝐽=∅, either 𝑐∈𝐼−𝐽 or 𝑐∈𝐽−𝐼 for all 𝑐∈𝐾. If 𝑐∈𝐼−𝐽, the ℎ_𝜇=𝑓_𝜇 is an ideal of 𝑅 since (𝑓,𝐼)⊳𝑅. If 𝑐∈𝐽−𝐼, then ℎ_𝜇=𝑔_𝜇 is an ideal of 𝑅 since (𝑔,𝐽)⊳𝑅. Thus, ℎ_𝜇 is an ideal of 𝑅 for all 𝑐∈𝐾, and so (𝑓,𝐼)∪(𝑔,𝐽)=(ℎ,𝐾)⊳𝑅.

Theorem 37. Let (𝐹,𝐼) be an intuitionistic fuzzy soft ideal over 𝑅. If {(𝐹_𝑗,𝐼_𝑗)∣𝑗∈𝐽} is a nonempty family of intuitionistic fuzzy soft ideals of (𝐹,𝐼), where 𝐽 is an index set, then,(1)∩_𝑗∈𝐽(𝐹_𝑗,𝐼_𝑗)⊳𝑅,(2)∧_𝑗∈𝐽(𝐹_𝑗,𝐼_𝑗)⊳𝑅,(3)∨_𝑗∈𝐽(𝐹_𝑗,𝐼_𝑗)⊳𝑅, where 𝐼_𝑗∩𝐼_𝑘={0} for all 𝑗,𝑘∈𝐽.

Definition 38. Let 𝑓∶𝑅→𝑆 and 𝑔∶𝐴→𝐵 be two functions, where 𝐴 and 𝐵 are parameter sets for intuitionistic fuzzy soft sets 𝑅 and 𝑆, respectively. Then, the pair (𝑓,𝑔) is called an intuitionistic fuzzy soft function from 𝑅 to 𝑆.

Definition 39. Let (𝐹,𝐴) and (𝐺,𝐵) be two intuitionistic fuzzy soft rings over 𝑅 and 𝑆, respectively, let 𝑓∶𝑅→𝑆 be a homomorphism of rings, and let 𝑔∶𝐴→𝐵 be a mapping of sets. Then, we say that (𝑓,𝑔)∶(𝐹,𝐴)→(𝐺,𝐵) is an intuitionistic fuzzy soft homomorphism of intuitionistic fuzzy soft rings and define by 𝑓(𝐹,𝐴)=(𝐺,𝐵)𝑔, if the following conditions are satisfied: 𝑓𝜇_𝐹(𝜀)(𝑥)=𝜇_𝐺(𝜀)(𝑥𝑓𝑣)𝑔,_𝐹(𝜀)(𝑥)=𝑣_𝐺(𝜀)(𝑥)𝑔. (33)

Definition 40. If there exists an intuitionistic fuzzy soft ring homomorphism between (𝐹,𝐴) and (𝐺,𝐵), we say that (𝐹,𝐴) is intuitionistic fuzzy soft homomorphic to (𝐺,𝐵) and is denoted by (𝐹,𝐴)∼(𝐺,𝐵). Moreover, if 𝑓 is an isomorphism, then (𝐹,𝐴) is intuitionistic fuzzy soft isomorphic to (𝐺,𝐵), which is denoted by (𝐹,𝐴)≃(𝐺,𝐵).

Now, we show that the homomorphic image and preimage of an intuitionistic fuzzy soft ring are also intuitionistic fuzzy soft rings.

Definition 41. Let (𝐹,𝐴) and (𝐺,𝐵) be two intuitionistic fuzzy soft rings over 𝑅 and 𝑆. Let (𝑓,𝑔) be an intuitionistic fuzzy soft function from 𝑅 to𝑆.(1)The image of (𝐹,𝐴) under the intuitionistic fuzzy soft function (𝑓,𝑔) denoted by (𝑓,𝑔)(𝐹,𝐴) is the intuitionistic fuzzy soft ring over 𝑆 defined by (𝑓,𝑔)(𝐹,𝐴)=(𝑓(𝐹),𝑔(𝐴)), where 𝑓𝐹_𝑘(𝑠)=_𝑓(⋁_𝑟)=𝑠⋁_{𝑔(𝑎)=𝑘}𝜇_𝐹(𝜀)(𝑟)if𝑟∈𝑓⁻¹(𝑠)𝑓𝐹0otherwise,_𝑘(𝑠)=_𝑓(⋀_𝑟)=𝑠⋀_{𝑔(𝑎)=𝑘}𝑣_𝐹(𝜀)(𝑥)if𝑟∈𝑓⁻¹(𝑠)1otherwise, (34)for all 𝑘∈𝑔(𝐴) and for all 𝑠∈𝑆. (2)The preimage of (𝐺,𝐵) under the intuitionistic fuzzy soft function (𝑓,𝑔) denoted by (𝑓,𝑔)⁻¹(𝐺,𝐵) is the intuitionistic fuzzy soft ring over 𝑅 defined by (𝑓,𝑔)⁻¹(𝐺,𝐵)=(𝑓⁻¹(𝐺),𝑔⁻¹(𝐵)), where 𝑓⁻¹(𝐺)_𝑎(𝑟)=(𝐺)_𝑔(𝑎)(𝑓(𝑟)), for all 𝑎∈𝑔⁻¹(𝐵) and for all 𝑟∈𝑅.
If 𝑓 and𝑔 are injective (surjective), then (𝑓,𝑔) is said to be injective (surjective).

Definition 42. Let (𝑓,𝑔) be an intuitionistic fuzzy soft function from 𝑅 to 𝑆. If 𝑓 is a homomorphism function from 𝑅 to𝑆, then (𝑓,𝑔) is said to be intuitionistic fuzzy soft homomorphism. If 𝑓 is an isomorphism from 𝑅 to 𝑆 and 𝑔 is a one-to-one mapping from 𝐴 onto𝐵, then (𝑓,𝑔) is said to be an intuitionistic fuzzy soft isomorphism.

Theorem 43. Let (𝐹,𝐴) be an intuitionistic fuzzy soft ring over 𝑅 and (𝑓,𝑔) an intuitionistic fuzzy soft homomorphism from 𝑅 to 𝑆. Then, (𝑓,𝑔)(𝐹,𝐴) is an intuitionistic fuzzy soft ring over 𝑆.

Proof. The proof is straightforward.

Theorem 44. Let (𝐺,𝐵) be an intuitionistic fuzzy soft ring over 𝑆 and (𝑓,𝑔) be an intuitionistic fuzzy soft homomorphism from 𝑅 to 𝑆. Then, (𝑓,𝑔)⁻¹(𝐺,𝐵) is an intuitionistic fuzzy soft ring over 𝑅.

Proof. The proof is straightforward.

4. Conclusion

In this work the theoretical point of view of intuitionistic fuzzy soft sets in ring and ideal is discussed. The work is focused on intuitionistic fuzzy soft rings and fuzzy soft ideals. These concepts are basic structures for improvement of fuzzy soft set theory. One can extend this work by studying other algebraic structures.

Acknowledgment

The authors are highly grateful to the referees for their valuable comments and suggestions for improving the paper.