An approach to fuzzy multi-ideals of near rings

Abstract

In recent years, fuzzy multisets have become a subject of great interest for researchers and have been widely applied to algebraic structures including groups, rings, and many other algebraic structures. In this paper, we introduce the algebraic structure of fuzzy multisets as fuzzy multi-subnear rings (multi-ideals) of near rings. In this regard, we define different operations on fuzzy multi-ideals of near rings and we generalize some results known for fuzzy ideals of near rings to fuzzy multi-ideals of near rings.

Keywords

Near ring multiset fuzzy multiset fuzzy multi-ideal fuzzy multi-subnear ring

1 Introduction

Recent developments in various algebraic structures and the applications of those in different areas play an important role in Science and Technology. One of the best tools to study the non-linear algebraic systems is the theory of near rings. The interest in near rings and near-fields started at the beginning of the 20th century, when L. Dickson wanted to know whether the list of axioms for skew fields is redundant or not. He found in [15] that there do exist “near fields" which fulfill all axioms for skew fields except one distributive law. Since 1950, the theory of near rings had applications to several domains, for instance in the area of dynamical systems, graphs, homological algebra, universal algebra, category theory, geometry, and so on. A near ring is an algebraic structure similar to a ring but satisfying fewer axioms. Although near rings arise naturally in various ways, most near rings studied today arise as the endomorphisms of a group or cogroup object of a category. For details about near ring theory and applications, we refer to [12 , 25]. For hyperrings and hyperideals we refer to e.g. [7 , 22].

A new mathematical structure, known by multiset or bag, was introduced by Yager in 1987. Yager [29] defined a multiset as a collection of elements (e.g., numbers or symbols) with the possibility that an element may occur more than once. The number of times an element occurs in a multiset is called its multiplicity in the multiset. A set is a multiset in which every element has multiplicity one i.e. its elements are pairwise different. Moreover, Zadeh [30], in 1965, proposed fuzzy sets as mathematical model of vagueness where elements belong to a given set to some degree that is typically a number that belongs to the unit interval [0, 1]. As a combination of the two concepts: multisets and fuzzy sets, Yager [29] defined fuzzy multisets or fuzzy bags. The latter are fuzzy subsets whose elements may occur more than one time. A similar generalization of fuzzy sets was found in 2014 where Chen et al. [11] introduced m-polar fuzzy sets as a map from a non-empty set X to [0, 1] ^m. The difference between fuzzy multisets and m-polar fuzzy sets is that in fuzzy multisets, the count function of an element is a decreasingly ordered sequence of numbers in [0, 1] whereas in m-polar fuzzy sets, it can be any sequence of m numbers in [0, 1].

Rosenfeld [24] first studied fuzzy subgroups of a group and Biswas [8] introduced anti-fuzzy subgroups and then the fuzzification of algebraic structures started to grow up. In particular, a link between near rings and fuzzy sets has been established by Abou-Zaid [1] where he discussed fuzzy subnear rings and ideals of near rings. Later on, other researchers considered this field of study such as Davvaz [14] who discussed fuzzy ideals of near rings with interval valued membership functions. Also, a book was written by Satyanarayana and Prasad on near rings and fuzzy ideals (see [25]). Some generalizations for fuzzy algebraic structures were found. In particular, a link between m-polar fuzzy sets and algebraic structures was found and some papers in this regard were published. For more details, we refer to [2 –4]. Moreover, a link between fuzzy multisets and algebraic structures was recently established and some work was done in this regard, see [5 , 23].

In [19], Mayerova and Al-Tahan highlighted the connection between fuzzy multisets and algebraic structures from an anti-fuzzification point of view by studying anti-fuzzy multi-ideals of near rings. In our paper and inspired by the previous work on fuzzy (multi-) algberaic structures, we highlight the connection between fuzzy multisets and near rings from a fuzzification point of view by studying fuzzy multi-subnear rings (multi-ideals) of near rings. The remainder of this paper is organized as follows: Section 2 briefly reviews some preliminary results related to near rings and fuzzy multisets that are used throughout the paper. Section 3 defines fuzzy multi-subnear rings (multi-ideals) of near rings and presents several results related to the new defined concepts. The obtained results are a general form of those related to fuzzy subsets of near rings and for fuzzy multi-subnear ring (multi-ideals) of rings as well.

2 Preliminaries

This section covers preliminary results related to near rings (see [12 , 28]) and fuzzy multisets (see [20 , 29]) that are used throughout the paper.

2.1 Near rings

A near ring is an algebraic structure that looks like a ring where it allows one distributive law to be satisfied (either left or right).

Definition 2.1. [12] Let R be a non-empty set. Then (R, + , ·) is called a near ring if:

(R, +) is a group;

(R, ·) is a semigroup;

x · (y + z) = x · y + x · z for all x, y, z ∈ R (Left distributive law).

Remark 1. Every ring is a near ring.

We present some examples of near rings that are not rings. More precisely, they do not satisfy the right distributive law.

Example 1. Let n be a natural number and $ℤ_{n}$ be the set of integers modulo n. Then $(ℤ_{n}, +, \cdot)$ is a near ring. Here “+" is the standard addition modulo n and “·" is defined as a · b = b for all $a, b \in ℤ_{n}$ .

Example 2. Let $ℤ$ be the set of integers. Then $(ℤ, +, \cdot)$ is a near ring. Here “+" is the standard addition and “·" is defined as a · b = b for all $a, b \in ℤ$ .

The following are examples on matrix near rings and polynomial near rings that differ from the known matrix rings and polynomial rings.

Example 3. Let n be a natural number, (R, + , ·) be a near ring with a · b = b for all a, b ∈ R and M_n (R) be the set of n × n matrices with entries from R. Then (M_n (R) , + , ·) is a near ring and it is called matrix near ring. Here, for all (a_ij) , (b_ij) ∈ M_n (R), (a_ij) + (b_ij) = (a_ij + b_ij) and (a_ij) · (b_ij) = (c_ij) where (c_ij) = $(\begin{matrix} b_{11} + b_{21} + \dots + b_{n 1} & b_{12} + b_{22} + \dots + b_{n 2} & \dots & b_{1 n} + b_{2 n} + \dots + b_{nn} \\ b_{11} + b_{21} + \dots + b_{n 1} & b_{12} + b_{22} + \dots + b_{n 2} & \dots & b_{1 n} + b_{2 n} + \dots + b_{nn} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ b_{11} + b_{21} + \dots + b_{n 1} & b_{12} + b_{22} + \dots + b_{n 2} & \dots & b_{1 n} + b_{2 n} + \dots + b_{nn} \end{matrix}) .$

Example 4. Let n be a natural number, (R, + , ·) be a near ring with a · b = b for all a, b ∈ R, and P_n (R) be the set of all polynomials with degree less than or equal to n and with coefficients from R. Then (P_n (R) , + , ·) is a near ring and it is called polynomial near ring. Where for all p (x) = a₀ + a₁x + … + a_nxⁿ, q (x) = b₀ + b₁x + … + b_nxⁿ ∈ P_n (R), p (x) + q (x) = (a₀ + b₀) + (a₁ + b₁) x + … + (a_n + b_n) xⁿ and p (x) · q (x) = q (x).

Definition 2.2. [12] Let (R, + , ·) be a near ring. A non-empty subset S of R is called a subnear ring of R if (S, + , ·) is a near ring.

Remark 2. To prove that (S, + , ·) is a subnear ring of (R, + , ·), it suffices to show that x - y ∈ S and x · y ∈ S for all x, y ∈ S.

Definition 2.3. [12] Let (R, + , ·) be a near ring. A subnear ring I of R is called an ideal of R if:

x + y - x ∈ I for all x ∈ R and y ∈ I;

x · y ∈ I for all x ∈ R and y ∈ I;

(x + a) · y - x · y ∈ I for all x, y ∈ R and a ∈ I.

Remark 3. Every near ring R ≠ {0} has at least two ideals: R and the trivial ideal {0}.

Example 5. Let $(ℤ, +, \cdot)$ be the near ring defined in Example 2. Then $(n ℤ, +, \cdot)$ , with $n \in ℤ$ , are the only ideals of $(ℤ, +, \cdot)$ .

Definition 2.4. [12] A near ring (R, + , ·) is called simple if R has no proper non-trivial ideals.

Example 6. Let $(ℤ_{n}, +, \cdot)$ be the near ring defined in Example 1. Then $(ℤ_{n}, +, \cdot)$ is simple if and only if n is a prime number.

2.2 Fuzzy multisets

A multiset (or bag) is a set containing repeated elements. A fuzzy multiset is a generalization of fuzzy set and it was introduced by Yager in [29] under the name fuzzy bag. In these fuzzy bags, the count of the number of elements itself becomes a crisp bag.

Definition 2.5. [30] Let U be any non-empty set. A fuzzy set on U is characterized by a membership function μ_A (x) that assigns any element in U a grade of membership in A. The fuzzy set may be represented by the set of ordered pairs A = {(x, μ_A (x)) : x ∈ U} , where μ_A (x) ∈ [0, 1].

Definition 2.6. [26] Let X be a non-empty set and Q be the set of all crisp multisets drawn from the interval [0, 1]. A fuzzy multiset A drawn from X is represented by a function CM_A : X → Q.

In the above definition, the value CM_A (x) is a crisp multiset drawn from [0, 1]. For each x ∈ X, CM_A (x) is defined as the decreasingly ordered sequence of elements and it is denoted by:

$(μ_{A}^{1} (x), μ_{A}^{2} (x), \dots, μ_{A}^{p} (x)) :$ $μ_{A}^{1} (x) \geq μ_{A}^{2} (x) \geq \dots \geq μ_{A}^{p} (x) .$ A fuzzy set on a set X can be considered as a special case of fuzzy multiset where $CM (x) = (μ_{A}^{1} (x))$ for all x ∈ X.

Example 7. If X = {a, b, c, d}, then $\begin{matrix} A = {(0.87, 0.45) / a, (0.67, 0.21, 0.2, 0.1) / c, \\ (0.34, 0.1) / d} and \\ B = {(1, 1) / a, (0.7, 0.6, 0.5, 0.1) / b, (0.3, 0.1) / c, \\ (0.5, 0.4, 0.1) / d} \end{matrix}$

are fuzzy multisets of X.

The fuzzy multisets A, B with fuzzy count functions CM_A, CM_B respectively can be written as: ${CM}_{A} (x) = {\begin{matrix} (0.87, 0.45) & if x = a, \\ 0 & if x = b, \\ (0.67, 0.21, 0.2, 0.1) & if x = c, \\ (0.34, 0.1) & if x = d . \end{matrix}$ and ${CM}_{B} (x) = {\begin{matrix} (1, 1) & if x = a, \\ (0.7, 0.6, 0.5, 0.1) & if x = b, \\ (0.3, 0.1) & if x = c, \\ (0.5, 0.4, 0.1) & if x = d . \end{matrix}$

Definition 2.7. [27] Let X be a non-empty set and A, B be fuzzy multisets of X with fuzzy count functions CM_A, CM_B respectively. Then:

A ⊆ B if CM_A (x) ≤ CM_B (x) for all x ∈ X;

A = B if CM_A (x) = CM_B (x) for all x ∈ X (i.e., CM_A (x) ≤ CM_B (x) and CM_B (x) ≤ CM_A (x) for all x ∈ X);

the fuzzy multiset A ∩ B is defined as CM_A∩B (x) = CM_A (x) ∧ CM_B (x);

the fuzzy multiset A ∪ B is defined as CM_A∪B (x) = CM_A (x) ∨ CM_B (x).

In Definition 2.7, if ${CM}_{A} (x) = (μ_{A}^{1} (x), \dots, μ_{A}^{p} (x))$ and ${CM}_{B} (x) = (μ_{B}^{1} (x), \dots, μ_{B}^{r} (x))$ then:

CM_A (x) ≤ CM_B (x) means that $μ_{A}^{i} (x) \leq μ_{B}^{i} (x)$ for all i = 1, …, max {p, r};

${CM}_{A} (x) \land {CM}_{B} (x) = (min (μ_{A}^{1} (x), μ_{B}^{1} (x)),$ $\dots, min (μ_{A}^{max (p, r)} (x), μ_{B}^{max (p, r)} (x)))$ ;

${CM}_{A} (x) \lor {CM}_{B} (x) = (max (μ_{A}^{1} (x), μ_{B}^{1} (x)),$ $\dots, max (μ_{A}^{max (p, r)} (x), μ_{B}^{max (p, r)} (x)))$ .

Example 8. Let X = {a, b, c, d} and A, B be the fuzzy multisets over X in Example 7. Then $A \cap B = {(0.87, 0.45) / a, (0.3, 0.1) / c, (0.34, 0.1) / d}$ and

$\begin{array}{l} A \cup B = {(1, 1) / a, (0.7, 0.6, 0.5, 0.1) / b, \\ (0.67, 0.21, 0.2, 0.1) / c, (0.5, 0.4, 0.1) / d} . \end{array}$

3 Fuzzy multi-ideal of near rings

This section combines the notions of fuzzy multiset and fuzzy ideals of near rings to define fuzzy multi-ideals of near rings. It presents several results related to the new defined concept. Since every fuzzy set can be considered as a fuzzy multiset, it follows that some of the results in this section can be considered as a generalization for those in [1] that are related to fuzzy ideals of near rings.

Definition 3.1. Let (R, + , ·) be a near ring. A fuzzy multiset A (with fuzzy count function CM_A) over R is a fuzzy multi-subnear ring of R if for all x, y ∈ R, the following conditions hold.

CM_A (x) ∧ CM_A (y) ≤ CM_A (x - y);

CM_A (x) ∧ CM_A (y) ≤ CM_A (x · y).

Definition 3.2. Let (R, + , ·) be a near ring. A fuzzy multiset A (with fuzzy count function CM_A) over R is a fuzzy multi-ideal of R if for all x, y ∈ R, the following conditions hold.

CM_A (x) ∧ CM_A (y) ≤ CM_A (x - y);

CM_A (x) ∧ CM_A (y) ≤ CM_A (x · y);

CM_A (y) ≤ CM_A (x + y - x);

CM_A (y) ≤ CM_A (x · y);

CM_A (a) ≤ CM_A ((x + a) y - xy) for all a ∈ R.

Remark 4. Let (R, + , ·) be a near ring, (R, +) an abelian group, and A a fuzzy multiset over R. To prove that A is a fuzzy multi-ideal of R, it suffices to show that 1., 4., and 5. of Definition 3.2 are satisfied.

Example 9. Let (R, + , ·) be a near ring with zero element 0 ∈ R and A be a fuzzy multiset of R defined as CM_A (x) = CM_A (0) for all x ∈ R. Then A is a fuzzy multi–ideal of R (the constant fuzzy multi-ideal).

Proposition 3.3. Let (R, + , ·) be a near ring with zero element 0 ∈ R and A be a fuzzy multi-ideal of R. Then

CM_A (x) ≤ CM_A (0);

CM_A (- x) = CM_A (x);

CM_A (x + y - x) = CM_A (y);

CM_A (a) ≤ CM_A (x) for all x ∈ < a > = {r · a : r ∈ R}.,

⋀_1≤i≤nCM_A (x_i) ≤ CM_A (x₁ + … + x_n) for all x_i ∈ R.

Proof. The proof is straightforward. □

Remark 5. Let (R, + , ·) be a near ring. Then we can define at least one fuzzy multi-ideal of R which is mainly the one that is defined in Example 9.

We present some examples on near rings with non-constant fuzzy multi-ideals.

Example 10. Let (R, + , ·) be the near ring defined by the following tables:

$\begin{matrix} + & 0 & 1 & 2 \\ 0 & 0 & 1 & 2 \\ 1 & 1 & 2 & 0 \\ 2 & 2 & 0 & 1 \end{matrix}$

$\begin{matrix} \cdot & 0 & 1 & 2 \\ 0 & 0 & 1 & 2 \\ 1 & 0 & 1 & 2 \\ 2 & 0 & 1 & 2 \end{matrix}$

It is clear that A = {(1, 1, 0.6, 0.6, 0.1)/0, (0.8, 0.4, 0.2, 0.1)/1, (0.8, 0.4, 0.2, 0.1)/2} is a fuzzy multi-ideal of R. Moreover, $B = {(1, 0.6, 0.6, 0.1) / 0, (0.8, 0.7, 0.1) / 1, (0.8, 0.7, 0.1) / 2}$ is not a fuzzy multi-ideal of R as CM_B (0) ≱CM_B (1) (By Proposition 3.3).

Example 11. Let $(ℤ, +, \cdot)$ be the near ring of integers under standard addition “+" and with “·" defined by x · y = y for all $x, y \in ℤ$ . Then A, B are fuzzy multi-ideals of $ℤ$ with the following fuzzy count functions respectively: ${CM}_{A} (x) = {\begin{matrix} (1, 1, 1, 1) if x is a multiple of 3; \\ (0.7, 0.6, 0.6) otherwise . \end{matrix}$ ${CM}_{B} (x) = {\begin{matrix} (0.9, 0.8, 0.7) if x is an even integer; \\ (0.8, 0.7, 0.7) otherwise . \end{matrix}$

Next, we deal with some operations on fuzzy multi-subnear rings (multi-ideals) of near rings such as intersection, union, and product.

Proposition 3.4. Let (R, + , ·) be a near ring and A, B be fuzzy multi-subnear rings of R. Then A ∩ B is a fuzzy multi-subnear ring of R.

Proof. Let x, y ∈ R. We need to show that conditions 1. and 2. of Definition 3.1 are satisfied for A ∩ B. Having CM_A∩B (x - y) = CM_A (x - y) ∧ CM_B (x - y) and A, B fuzzy multi-subnear rings of R implies that CM_A (x - y) ≥ CM_A (x) ∧ CM_A (y) and CM_B (x - y) ≥ CM_B (x) ∧ CM_B (y). The latter implies that CM_A∩B (x - y) ≥ CM_A (x) ∧ CM_A (y) ∧ CM_B (x) ∧ CM_B (y) = CM_A∩B (x) ∧ CM_A∩B (y). Moreover, having CM_A∩B (x · y) = CM_A (x · y) ∧ CM_B (x · y) and A, B fuzzy multi-subnear rings of R implies that CM_A (x · y) ≥ CM_A (x) ∧ CM_A (y) and CM_B (x · y) ≥ CM_B (x) ∧ CM_B (y). The latter implies that CM_A∩B (x · y) ≥ CM_A (x) ∧ CM_A (y) ∧ CM_B (x) ∧ CM_B (y) = CM_A∩B (x) ∧ CM_A∩B (y). □

Corollary 3.5. Let (R, + , ·) be a near ring and A_i be a fuzzy multi-subnear ring of R for i = 1, …, n. Then $⋂_{i = 1}^{n} A_{i}$ is a fuzzy multi-subnear ring of R.

Proof. The proof follows from Proposition 3.4 and by using induction. □

Proposition 3.6. Let (R, + , ·) be a near ring and A, B be fuzzy multi-ideals of R. Then A ∩ B is a fuzzy multi-ideal of R.

Proof. Let x, y ∈ R. We need to show that conditions of Definition 3.2 are satisfied for A ∩ B. Proposition 3.4 asserts that Conditions 1. and 2. of Definition 3.2 are satisfied. We need to show that Condition 3., 4., and 5. are satisfied. Having CM_A∩B (x + y - x) = CM_A (x + y - x) ∧ CM_B (x + y - x) and A, B fuzzy multi-ideals of R implies that CM_A (x + y - x) ≥ CM_A (y) and CM_B (x + y - x) ≥ CM_B (y). The latter implies that CM_A∩B (x + y - x) ≥ CM_A (y) ∧ CM_B (y) = CM_A∩B (y). Moreover, having CM_A∩B (x · y) = CM_A (x · y) ∧ CM_B (x · y) and A, B fuzzy multi-ideals of R implies that CM_A (x · y) ≥ CM_A (y) and CM_B (x · y) ≥ CM_B (y). The latter implies that CM_A∩B (x · y) ≥ CM_A (y) ∧ CM_B (y) = CM_A∩B (y). Let a ∈ R. Having CM_A∩B ((x + a) y - xy) = CM_A ((x + a) y - xy) ∧ CM_B ((x + a) y - xy) and A, B fuzzy multi-ideals of R implies that CM_A ((x + a) y - xy) ≥ CM_A (a) and CM_B ((x + a) y - xy) ≥ CM_B (a). The latter implies that CM_A∩B ((x + a) y - xy) ≥ CM_A (a) ∧ CM_B (a) = CM_A∩B (a).□

Corollary 3.7. Let (R, + , ·) be a near ring and A_i be a fuzzy multi-ideal of R for i = 1, …, n. Then $⋂_{i = 1}^{n} A_{i}$ is a fuzzy multi-ideal of R.

Proof. The proof follows from Proposition 3.6 and by using induction.□

Example 12. Let $(ℤ, +, \cdot)$ be the near ring of integers defined in Example 11 and A, B be the fuzzy multi-ideals of $ℤ$ given by: ${CM}_{A} (x) = {\begin{matrix} (0.8, 0.5, 0.5) if x is a multiple of 3; \\ 0 otherwise . \end{matrix}$ and $\begin{matrix} {CM}_{B} (x) \\ = {\begin{matrix} (0.7, 0.6, 0.6, 0.6) if x is an even integer; \\ 0 otherwise . \end{matrix} \end{matrix}$

One can easily see that the fuzzy multi-ideal A ∩ B of $ℤ$ is given as follows: ${CM}_{A \cap B} (x) = {\begin{matrix} (0.7, 0.5, 0.5) if x is a multiple of 6; \\ 0 otherwise . \end{matrix}$ Remark 6. Let A, B be fuzzy multi-ideals of a near ring R. In some cases like when one of them is the constant fuzzy ideal or one of them is a subset of the other, A ∪ B is a fuzzy multi-ideal of R. But in general, A ∪ B may not be a fuzzy multi-ideal of R. We illustrate Remark 6 by the following example.

Example 13. Let $(ℤ, +, \cdot)$ be the near ring of integers defined in Example 11 and A, B be the fuzzy multi-ideals of $ℤ$ given by: ${CM}_{A} (x) = {\begin{matrix} (1, 1, 1, 1) if x is a multiple of 3; \\ (0.7, 0.6, 0.6) otherwise . \end{matrix}$ ${CM}_{B} (x) = {\begin{matrix} (0.9, 0.8, 0.7) if x is an even integer; \\ (0.8, 0.7, 0.7) otherwise . \end{matrix}$

Having CM_A∪B (3) = (1, 1, 1, 1) , CM_A∪B (2) = (0.9, 0.8, 0.7) , CM_A∪B (1) = (0.8, 0.7, 0.7), 1 = 3 -2, and CM_A∪B (1) = (0.8, 0.7, 0.7) ≱CM_A∪B (3) ∧ CM_A∪B (2) = (0.9, 0.8, 0.7) implies that A ∪ B is not a fuzzy multi-ideal of $ℤ$ .

Proposition 3.8. Let R, S be near rings with fuzzy multisets A, B respectively. If A, B are fuzzy multi-subnear rings (multi-ideals) of R, S then A × B is a fuzzy multi-subnear rings (multi-ideal) of R × S, where CM_A×B ((r, s)) = CM_A (r) ∧ CM_B (s) for all (r, s) ∈ R × S.

Proof. The proof is straightforward.□

Proposition 3.9. Let R_i be near rings with fuzzy multiset A_i for i = 1, …, n. If A_i is a fuzzy multi-subnear ring (multi-ideal) of R_i then $\prod_{i = 1}^{n} A_{i}$ is a fuzzy multi-ideal of $\prod_{i = 1}^{n} R_{i}$ , where ${CM}_{\prod_{i = 1}^{n} A_{i}} ((r_{1}, \dots, r_{n})) = ⋀_{i = 1}^{n} {CM}_{A_{i}} (r_{i})$ for all $(r_{1}, \dots, r_{n}) \in \prod_{i = 1}^{n} R_{i}$ .

Proof. The proof follows from Proposition 3.8 and by using induction.□

Example 14. Let (R, + , ·) be the near ring in Example 10 and (S, + ₁, · ₁) be the near ring, with a as a zero element, defined by the following tables: $\begin{matrix} +_{1} & a & b \\ a & a & b \\ b & b & a \end{matrix}$ and $\begin{matrix} \cdot_{1} & a & b \\ a & a & b \\ b & a & b \end{matrix}$

Then the near ring (R × S, ⊕ , ⊗) is given by the following tables: $\begin{matrix} \oplus & (0, a) & (0, b) & (1, a) & (1, b) & (2, a) & (2, b) \\ (0, a) & (0, a) & (0, b) & (1, a) & (1, b) & (2, a) & (2, b) \\ (0, b) & (0, b) & (0, a) & (1, b) & (1, a) & (2, b) & (2, a) \\ (1, a) & (1, a) & (1, b) & (2, a) & (2, b) & (0, a) & (0, b) \\ (1, b) & (1, b) & (1, a) & (2, b) & (2, a) & (0, b) & (0, a) \\ (2, a) & (2, a) & (2, b) & (0, a) & (0, b) & (1, a) & (1, b) \\ (2, b) & (2, b) & (2, a) & (0, b) & (0, a) & (1, b) & (1, a) \end{matrix}$

$\begin{matrix} \otimes & (0, a) & (0, b) & (1, a) & (1, b) & (2, a) & (2, b) \\ (0, a) & (0, a) & (0, b) & (1, a) & (1, b) & (2, a) & (2, b) \\ (0, b) & (0, a) & (0, b) & (1, a) & (1, b) & (2, a) & (2, b) \\ (1, a) & (0, a) & (0, b) & (1, a) & (1, b) & (2, a) & (2, b) \\ (1, b) & (0, a) & (0, b) & (1, a) & (1, b) & (2, a) & (2, b) \\ (2, a) & (0, a) & (0, b) & (1, a) & (1, b) & (2, a) & (2, b) \\ (2, b) & (0, a) & (0, b) & (1, a) & (1, b) & (2, a) & (2, b) \end{matrix}$

It is clear that A = {(0.7, 0.6, 0.6, 0.5)/0, (0.6, 0.5, 0.5, 0.5)/1, (0.6, 0.5, 0.5, 0.5)/2} and B = {(0.6, 0.6, 0.6, 0.6, 0.6)/a, (0.6, 0.6, 0.6, 0.6, 0.6)/b} are fuzzy multi-ideals of R and S, respectively. Using Proposition 3.8, we get that A × B is a fuzzy multi-ideal of R × S, where ${CM}_{A \times B} (x, y) = {\begin{matrix} (0.6, 0.6, 0.6, 0.5) if x = 0; \\ (0.6, 0.5, 0.5, 0.5) otherwise . \end{matrix}$

Notation 1. Let (R, + , ·) be a near ring, A be a fuzzy multiset of R and ${CM}_{A} (x) = (μ_{A}^{1} (x), μ_{A}^{2} (x),$ $\dots, μ_{A}^{p} (x))$ . Then

CM_A (x) =0 if $μ_{A}^{1} (x) = 0$ ,

CM_A (x) >0 if $μ_{A}^{1} (x) > 0$ ,

${CM}_{A} (x) = \underline{1}$ if ${CM}_{A} (x) = (\underset{s times}{\underset{︸}{1, \dots, 1}})$ where $s = max {k \in ℕ : {CM}_{A} (y) = (μ_{A}^{1} (y), μ_{A}^{2} (y),$ $\dots, μ_{A}^{k} (y)), μ_{A}^{k} (y) \neq 0, y \in R} .$

Definition 3.10. Let (R, + , ·) be a near ring and A be a fuzzy multiset of R. Then A_★ = {x ∈ R : CM_A (x) >0} and A^★ = {x ∈ R : CM_A (x) = CM_A (0)}. Here, A_★ is called the support of A.

Example 15. Let $ℤ$ be the near ring of integers (Example 1) and A, B be the fuzzy multi-ideals of $ℤ$ defined in Example 11. Then $A_{★} = B_{★} = ℤ$ , $A^{★} = 3 ℤ$ , and $B^{★} = 2 ℤ$ .

Proposition 3.11. Let (R, + , ·) be a near ring and A be a fuzzy multi-subnear ring (multi-ideal) of R. Then A_★ is either the empty set or a subnear ring (ideal) of R.

Proof. Let x, y∈ A_★ ≠ ∅. Having A a fuzzy multi-ideal of R implies that conditions of Definition 3.2 are satisfied. Condition 1. implies that x - y ∈ A_★ as CM_A (x - y) ≥ CM_A (x) ∧ CM_A (y) >0. Condition 2. implies that x · y ∈ A_★ as CM_A (x · y) ≥ CM_A (x) ∧ CM_A (y) >0. Condition 3. implies that r + y - r ∈ A_★ for all r ∈ R as CM_A (r + y - r) ≥ CM_A (y) >0. Condition 4. implies that r · y ∈ A_★ for all r ∈ R as CM_A (r · y) ≥ CM_A (y) >0. Condition 5 implies that (r + a) s - rs ∈ A_★ for all a ∈ A_★ and r, s ∈ R as CM_A ((r + a) s - rs) ≥ CM_A (a) >0.□

Proposition 3.12. Let (R, + , ·) be a near ring and A be a fuzzy multi-subnear ring (multi-ideal) of R. Then A^★ is a subnear ring (ideal) of R.

Proof. Having 0 ∈ A^★ implies that A^★≠ ∅. Let x, y∈ A^★ ≠ ∅. Having A a fuzzy multi-ideal of R implies that conditions of Definition 3.2 are satisfied. Having CM_A (x - y) ≥ CM_A (x) ∧ CM_A (y) = CM_A (0) and that CM_A (x) ≤ CM_A (0) for all x ∈ R implies that CM_A (x - y) = CM_A (0) and hence, x - y ∈ A^★. Having CM_A (x · y) ≥ CM_A (x) ∧ CM_A (y) = CM_A (0) and that CM_A (x) ≤ CM_A (0) for all x ∈ R implies that CM_A (x · y) = CM_A (0) and hence, x · y ∈ A^★. Having CM_A (r + y - r) ≥ CM_A (y) = CM_A (0) and that CM_A (x) ≤ CM_A (0) for all x ∈ R implies that CM_A (r + y - r) = CM_A (0) and hence, r + y - r ∈ A^★ for all r ∈ R. Having CM_A (r · y) ≥ CM_A (y) = CM_A (0) and that CM_A (x) ≤ CM_A (0) for all x ∈ R implies that CM_A (r · y) = CM_A (0) and hence, r · y ∈ A^★ for all r ∈ R. Having CM_A ((r + a) s - rs) ≥ CM_A (a) = CM_A (0) for all a ∈ A^★ and r, s ∈ R and that CM_A (x) ≤ CM_A (0) for all x ∈ R implies that CM_A ((r + a) s - rs) = CM_A (0) and hence, (r + a) s - rs ∈ A^★.□

Proposition 3.13. Let (R, + , ·) be a near ring with a · b = b for all a, b ∈ R and A be a fuzzy multi-ideal of R with fuzzy count function CM_A. Then CM is a fuzzy multi-ideal of the matrix near ring M_n (R), where CM ((a_ij)) = ⋀ _1≤i,j≤nCM_A (a_ij) for all a_ij ∈ M_n (R).

Proof. Let (a_ij) , (b_ij) ∈ M_n (R). We need to show that the conditions of Definition 3.2 are satisfied for CM.

(1) $\begin{matrix} CM ((a_{ij}) - (b_{ij})) = \\ = \land_{1 \leq i, j \leq n} {CM}_{A} (a_{ij} - b_{ij}) \geq \\ \land_{1 \leq i, j \leq n} ({CM}_{A} (a_{ij}) \land {CM}_{A} (b_{ij})) = \\ = (\land_{1 \leq i, j \leq n} {CM}_{A} (a_{ij})) \land (\land_{1 \leq i, j \leq n} {CM}_{A} (b_{ij})) \\ = CM ((a_{ij})) \land CM ((b_{ij})) . \end{matrix}$

(2) and (4)

Having (a_ij) (b_ij) = (c_ij) where c_ij = b_1j + … + b_nj implies that CM ((a_ij) (b_ij) ) = CM ((c_ij)) = ∧ _1≤i,j≤nCM_A (c_ij) = ∧ _1≤i,j≤nCM_A (c_1j).

But CM_A (c_1j) ≥ CM_A (b_1j) ∧ … ∧ CM_A (b_nj). Thus, CM ((a_ij) (b_ij) ) ≥ ∧ _1≤i,j≤n (CM_A (b_1j ) ∧ … ∧ CM_A (b_nj)) = ∧ _1≤i,j≤nCM_A (b_ij) = CM ((b_ij) ).

(3)

CM ((a_ij) + (b_ij) - (a_ij) ) = ∧ _1≤i,j≤nCM_A (a_ij + b_ij - a_ij) ≥ ∧ _1≤i,j≤nCM_A (b_ij) = CM ((b_ij)).

(5)

Let (c_ij) ∈ M_n (R). Having ((a_ij) + (b_ij)) (c_ij) - (a_ij) (c_ij) = (d_ij) where d_ij = 0 for all ≤i, j ≤ n implies CM (((a_ij) + (b_ij)) (c_ij) - (a_ij) (c_ij)) = CM ((d_ij)) = ∧ _1≤i,j≤nCM_A (d_ij) = CM_A (0) ≥ CM_A (b_ij) ≥ CM ((b_ij) ).□

Example 16. Let $(ℤ, +, \cdot)$ be the near ring with standard addition and x · y = y for all $x, y \in ℤ$ . Having A a fuzzy multi-ideal of $ℤ$ with ${CM}_{A} (x) = {\begin{matrix} (0.8, 0.5, 0.5) if x is a multiple of 3; \\ 0 otherwise . \end{matrix}$ implies that

$CM ((\begin{matrix} a & b \\ c & d \end{matrix})) =$ $= {\begin{matrix} (0.8, 0.5, 0.5) if a, b, c, d are multiples of 3; \\ 0 otherwise . \end{matrix}$ is a fuzzy multi-ideal of $M_{2} (ℤ)$ .

Proposition 3.14. Let (R, + , ·) be a near ring and (P_n (R) , + , ·) be the polynomial near ring defined in Example 4. If A is a fuzzy multi-ideal R then the fuzzy multiset of P_n (R) defined as: CM (a₀ + a₁x + … + a_nxⁿ) = ⋀ _0≤i≤nCM_A (a_i) is a fuzzy multi-ideal of P_n (R).

Proof. Let p (x) = a₀ + a₁x + … + a_nxⁿ, q (x) = b₀ + b₁x + … + b_nxⁿ, r (x) = r₀ + r₁x + … + r_nxⁿ ∈ P_n (R). We need to show that conditions of Definition 3.2 are satisfied for CM. (1) CM (p (x) - q (x)) = ⋀ _0≤i≤nCM_A (a_i - b_i). Having A a fuzzy multi-ideal of R implies that CM_A (a_i - b_i) ≥ CM_A (a_i) ∧ CM_A (b_i).

Thus, CM (p (x) - q (x) ) ≥ ⋀ _0≤i≤n (CM_A (a_i) ∧ CM_A (b_i) ) = (⋀ _0≤i≤nCM_A (a_i) ) newline ∧ (⋀ _0≤i≤nCM_A (b_i) ) = CM (p (x)) ∧ CM (q (x)).

(2) and (4)

Having p (x) · q (x) = q (x) implies the CM (p (x) · q (x)) = CM (q (x)) ≥ CM (q (x)).

(3)

Having CM (p (x) + q (x) - p (x)) = ⋀ _0≤i≤nCM_A (a_i + b_i - a_i) and A a fuzzy multi-ideal of R implies that CM_A (a_i + b_i - a_i) ≥ CM_A (b_i). The latter implies that CM (p (x) + q (x) - p (x)) ≥ ⋀ _0≤i≤nCM_A (b_i) = CM (q (x)).

(5)

Having 0 = (p (x) + r (x)) q (x) - p (x) q (x) and CM (0) = CM_A (0) ≥ CM_A (r_i) ≥ newline ⋀ _0≤i≤nCM_A (r_i) = CM (r (x)) implies that

CM ((p (x) + r (x)) q (x) - p (x) q (x) ) ≥ CM (r (x)).□

Example 17. Let (S, + ₁, · ₁) be the near ring defined in Example 14 with the fuzzy multi-ideal A = {(0.7, 0.7, 0.4)/a, (0.7, 0.4)/b}. Then

CM (a₀ + a₁x + a₂x²) = $= {\begin{matrix} (0.7, 0.7, 0.4) if a_{0} = a_{1} = a_{2} = a; \\ (0.7, 0.4) otherwise . \end{matrix}$ is a fuzzy multi-ideal of P₂ (S).

Proposition 3.15. Let (R, + , ·) be a near ring with a fuzzy multi-ideal A. Then the fuzzy multiset of P_n (R) defined as: CM (a₀ + a₁x + … + a_nx) = CM_A (a₀) is a fuzzy multi-ideal of P_n (R).

Proof. The proof is similar to that of Proposition 3.14.

□

Example 18. Let (S, + ₁, · ₁) be the near ring defined in Example 14 with the fuzzy multi-ideal A = {(0.7, 0.7, 0.4)/a, (0.7, 0.4)/b}. Then $CM (a_{0} + a_{1} x + a_{2} x^{2}) = {\begin{matrix} (0.7, 0.7, 0.4) if a_{0} = a; \\ (0.7, 0.4) otherwise . \end{matrix}$ is a fuzzy multi-ideal of P₂ (S).

Notation 2. Let (R, + , ·) be a near ring, A be a fuzzy multiset of R and ${CM}_{A} (x) = (μ_{A}^{1} (x), μ_{A}^{2} (x),$ $\dots, μ_{A}^{p} (x))$ . We say that CM_A (x) ≥ (t₁, …, t_k) if p ≥ k and $μ_{A}^{i} (x) \geq t_{i}$ for all i = 1, …, k. If CM_A (x) ≱ (t₁, …, t_k) and (t₁, …, t_k) ≱CM_A (x) then we say that CM_A (x) and (t₁, …, t_k) are not comparable.

Notation 3. Let (R, + , ·) be a near ring, A a fuzzy multiset of R with fuzzy count function CM, and t = (t₁, …, t_k) where t_i ∈ [0, 1] for i = 1, …, k and t₁ ≥ t₂ ≥ … ≥ t_k. Then CM_t = {x ∈ R : CM (x) ≥ t}.

Theorem 3.16. Let (R, + , ·) be a near ring, A a fuzzy multiset of R with fuzzy count function CM and t = (t₁, …, t_k) where t_i ∈ [0, 1] for i = 1, …, k and t₁ ≥ t₂ ≥ … ≥ t_k. Then A is a fuzzy multi-subnear ring of R if and only if CM_t is either the empty set or a subnear ring of R.

Proof. Let A be a fuzzy multi-subnear ring of R and x, y∈ CM_t ≠ ∅. We need to show that x - y ∈ CM_t and x · y ∈ CM_t. Since CM (x - y) ≥ CM (x) ∧ CM (y) ≥ t and CM (x · y) ≥ CM (x) ∧ CM (y) ≥ t, it follows that x - y ∈ CM_t and x · y ∈ CM_t. Thus, CM_t is a subnear ring of R. Conversely, let CM_t≠ ∅ be a subnear ring of R and x, y ∈ R. By setting t₀ = CM (x) ∧ CM (y), we get that x, y ∈ CM_{t
₀}. Having CM_{t
₀} a subnear ring of R implies that x - y ∈ CM_{t
₀} and x · y ∈ CM_{t
₀}. Thus, CM (x - y) ≥ t₀ = CM (x) ∧ CM (y) and CM (x · y) ≥ t₀ = CM (x) ∧ CM (y).□

Theorem 3.17. Let (R, + , ·) be a near ring, A a fuzzy multiset of R with fuzzy count function CM and t = (t₁, …, t_k) where t_i ∈ [0, 1] for i = 1, …, k and t₁ ≥ t₂ ≥ … ≥ t_k. Then A is a fuzzy multi-ideal of R if and only if CM_t is either the empty set or an ideal of R.

Proof. Let A be a fuzzy multi-ideal of R, r, s ∈ R and x, y∈ CM_t ≠ ∅. Theorem 3.16 asserts that x - y ∈ CM_t and x · y ∈ CM_t. We need to show that r + y - r ∈ CM_t, r · y ∈ CM_t, and (r + y) s - rs ∈ CM_t. Since CM (r + y - r) ≥ CM (y) ≥ t, CM (r · y) ≥ CM (y) ≥ t, and CM ((r + y) s - rs) ≥ CM (y) ≥ t, it follows that r + y - r ∈ CM_t, r · y ∈ CM_t, and (r + y) s - rs ∈ CM_t. Thus, CM_t is an ideal of R. Conversely, let CM_t≠ ∅ be an ideal of R and x, y, a ∈ R. Theorem 3.16 asserts that conditions 1. and 2. of Definition 3.2 are satisfied. We need to prove that conditions 3., 4., and 5. of Definition 3.2 are satisfied. (3) Let t₀ = CM (y). Then y ∈ CM_{t
₀}. Having CM_{t
₀} an ideal of R and y ∈ CM_{t
₀} implies that x + y - x ∈ CM_{t
₀}. The latter implies that CM (x + y - x) ≥ t₀ = CM (y).

(4) Having CM_{t
₀} an ideal of R and y ∈ CM_{t
₀} implies that x · y ∈ CM_{t
₀}. The latter implies that CM (x · y) ≥ t₀ = CM (y).

(5) Let t₁ = CM (a). Having CM_{t
₁} an ideal of R and a ∈ CM_{t
₁} implies that (x + a) y - xy ∈ CM_{t
₁}. The latter implies that CM ((x + a) y - xy) ≥ t₁ = CM (a).□

Example 19. Let $(M_{2} (ℤ), +, \cdot)$ be the matrix near ring defined in Example 16. Using Theorem 3.17, we get that ${CM}_{(0.8, 0.5, 0.5)} = M_{2} (3 (ℤ))$ is an ideal of $M_{2} (ℤ)$ .

Example 20. Let P₂ (S) = {a, b, bx, bx², b + bx, b + bx², bx + bx², b + bx + bx²} be the polynomial near ring in Example 18. Then, by using Theorem 3.17, we get that CM_{(0.7,0.7,0.4)} = {a, bx, bx², bx + bx²} is an ideal of P₂ (S).

Corollary 3.18. Let (R, + , ·) be a near ring. If every subnear ring of R is an ideal of R then every fuzzy multi-subnear ring of R is a fuzzy multi-ideal of R.

Proof. The proof follows from Theorems 3.16 and 3.17.□

Example 21. Let $(ℤ, +, \cdot)$ be the near ring defined in Example 11. Since every subnear ring of $ℤ$ is an ideal of $ℤ$ , it follows by Corollary 3.18 that every fuzzy multi-subnear ring of $ℤ$ is a fuzzy multi-ideal of $ℤ$ .

Remark 7. Every fuzzy multi-ideal of a near ring R is a fuzzy multi-subnear ring of R. But the converse may not always hold.

We present an example on a fuzzy multi-subnear ring of a near ring that is not a fuzzy multi-ideal.

Example 22. let (S₃, ∘) be the symmetric group on the set {1, 2, 3} where ∘ is the composition of permutations. We define ★ on S₃ as α ★ β = β for all α, β ∈ S₃. It is easy to see that (S₃, ∘ , ★) is a near ring and that {(1) , (12)} is a subnear ring of S₃ that is not an ideal. Using Theorems 3.16 and 3.17, we deduce that S₃ has a fuzzy multi-subnear ring that is not a fuzzy multi-ideal. More precisely, the following fuzzy multiset with count function CM is a fuzzy multi-subnear ring that is not a fuzzy multi-ideal. $CM (α) = {\begin{matrix} (0.8, 0.75, 0.75) if α = (1) or α = (12); \\ (0.65, 0.43) otherwise . \end{matrix}$ Proposition 3.19. Let (R, + , ·) be a near ring and I be an ideal of R. Then I = CM_t for some fuzzy multi-ideal CM of R, t = (t₁, …, t_k) where t_i ∈ [0, 1] for i = 1, …, k and t₁ ≥ t₂ ≥ … ≥ t_k.

Proof. Let I be an ideal of R, t = (t₁, …, t_k) where t_i ∈ [0, 1] for i = 1, …, k and define the fuzzy multiset A of R as follows: $CM (x) = {\begin{matrix} t if x \in I \\ 0 otherwise . \end{matrix}$ It is clear that I = CM_t. We still need to prove that CM is a fuzzy multi-ideal of R. Using Theorem 3.17, it suffices to show that CM_α≠ ∅ is an ideal of R for all α = (a₁, …, a_s) with a_i ∈ [0, 1] and a₁ ≥ … ≥ a_s for i = 1, …, s. One can easily see that

CM_α = $= {\begin{matrix} R if α = 0 \\ I if 0 < α \leq t \\ \emptyset if (α > t) or (α and t are not comparable) . \end{matrix}$ Thus, CM_α is either the empty set or an ideal of R.□

Corollary 3.20. Let (R, + , ·) be a near ring. Then R has at least one fuzzy multi-ideal beside the constant fuzzy multi-ideal.

Proof. Let (R, + , ·) be a near ring. Then R has at least two ideals: R and {0}. Having {0} an ideal of R and using Proposition 3.19, we get that {0} = CM_t for some t = (t₁, …, t_k). One can define the fuzzy multi-ideal corresponding to {0} as: $CM (x) = {\begin{matrix} t if x = 0 \\ q otherwise, \end{matrix}$ where q < t.□

Definition 3.21. Let (R, + , ·) be a near ring, A and B be fuzzy multi-subnear rings (ideals) of R with fuzzy count functions CM_A and CM_B respectively. Then A and B are equivalent fuzzy multi-subnear rings (ideals) of R if for all x, y ∈ R, the following conditions hold.

1. CM_A (x) = CM_A (y) if and only if CM_B (x) = CM_B (y),

2. CM_A (x) ≤ CM_A (y) if and only if CM_B (x) ≤ CM_B (y).

Example 23. Let (R, + , ·) be any non-trivial near ring with identity 0 ∈ R. Then A = {(0.8, 0.8, 0.7)/0} and B = {(1, 0.8, 0.7, 0.4, 0.3, 0.1, 0.1)/0} are equivalent fuzzy multi-ideals of R.

In what follows, when we say that a near ring has certain number of fuzzy multi-ideals we mean that it has a certain number of non-equivalent fuzzy multi-ideals.

Corollary 3.22. Let (R, + , ·) be a simple near ring. Then R has two fuzzy multi-ideals.

Proof. This is easily seen as R has only two ideals {0} and R. Corollary 3.20 completes the proof.□

Example 24. Let p be a prime number and $(ℤ_{p}, +, \cdot)$ be the near ring with standard addition modulo p and x · y = y for all $x, y \in ℤ_{p}$ . Then $ℤ_{p}$ has two fuzzy multi-ideals.

Corollary 3.23. Let R, S be simple near rings. Then R × S has six fuzzy multi-ideals.

Proof. It is easy to see that R × S has only four ideals: {0} × {0}, R × {0}, {0} × S, and R × S. The ideal {0} × {0} corresponds to the fuzzy multi-ideal ${CM}_{1} ((x, y)) = {\begin{matrix} t if (x, y) = (0, 0) \\ q otherwise, \end{matrix}$ where q < t.

The ideal R × {0} corresponds to the fuzzy multi-ideals ${CM}_{2} ((x, y)) = {\begin{matrix} t if y = 0 \\ q otherwise, \end{matrix}$ where q < t and ${CM}_{3} ((x, y)) = {\begin{matrix} t if x = y = 0 \\ q if x \neq 0, y = 0 \\ r otherwise, \end{matrix}$ where r < q < t. The ideal {0} × S corresponds to the fuzzy multi-ideal ${CM}_{4} ((x, y)) = {\begin{matrix} t if x = 0 \\ q otherwise, \end{matrix}$ where q < t and ${CM}_{5} ((x, y)) = {\begin{matrix} t if x = y = 0 \\ q if x = 0, y \neq 0 \\ r otherwise, \end{matrix}$ where r < q < t.

And the ideal R × S corresponds to the constant fuzzy multi-ideal of R × S.□

4 Conclusion

This paper found a new link between algebraic structures and fuzzy multisets by introducing fuzzy multi-ideals of near rings and studying its properties. The various basic operations, definitions, and theorems related to fuzzy multi-ideals of near rings have been discussed. The results in this paper can be considered as a generalization of the results known for fuzzy ideals of near rings. Also, our results are considered as a generalization for fuzzy ideals of rings. This is because every ring is a near ring.

For future research, we raise the following ideas.

Characterize fuzzy multisets of other algebraic structures such as fields, modules, vector spaces, etc.

Define generalized fuzzy multi-ideals of near rings and investigate their properties.

Footnotes

Acknowledgments

This research was supported by the grant VAROPS (DZRO FVT 3) granted by the Ministry of Defence of the Czech Republic.

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