Fuzzy multi-polygroups

Abstract

A multiset is a set containing repeated elements. The objective of this paper is to combine the innovative concept of multisets and polygroups. In particular, we use multisets and fuzzy multisets to introduce the concepts of multi-polygroups and fuzzy multi-polygroups respectively and we discuss their properties. Moreover, we construct fuzzy multi-polygroups from multi-polygroups.

Keywords

polygroup multiset fuzzy multiset multi-polygroup fuzzy multi-polygroup

1 Introduction

Algebraic hyperstructures represent a natural generalization of classical algebraic structures and they were introduced by Marty [20] in 1934 at the eighth Congress of Scandinavian Mathematicians. Where he generalized the notion of a group to that of a hypergroup. In a group, the composition of two elements is an element whereas in a hypergroup, the composition of two elements is a set. Hypergroups have been used in algebra, geometry, convexity, automata theory, combinatorial problems of coloring, lattice theory, Boolean algebras, logic etc., over the years. An overview of the most important works and results in the field of hyperstructures up to 1993 is given in the book of Corsini [10]. A further overview of the work can be found in the book of Corsini and Leoreanu [11] and on the website of Prof. Vougiouklis and his team of collaborators (see web page: http://aha.eled.duth.gr/Thesaurus1.1.htm) [33 –35]. Moreover, a comprehensive review was published in 2005 by Chvalina and Hoskova [18]. For recent similar overview of hyperstructures, we refer to [2 , 13–15]. Some interesting results concerning hyperstructures theory can be also find in recent papers e.g. [12 , 31]. Comer [7] introduced a special class of hypergroups, using the name of polygroups. He emphasized the importance of polygroups, by analyzing them in connections to graphs, relations, Boolean and cylindric algebras.

A set is a well-defined collection of distinct objects, that is, the elements of a set are pairwise different. A generalization of the notion of the set was introduced by Yager [36]. Yager introduced the bag (multiset) structure as a set-like object in which repeated elements are significant. He discussed operations on multisets such as intersection and showed the usefulness of the new defined structure in relational databases. Multisets are useful structures and they have found numerous applications in mathematics and computer science [21, 26]. For example, the prime factorization of an integer n > 0 is a multiset N whose elements are primes (e.g. 45 has the multiset {3, 3, 5}). Moreover, the eigenvalues of a matrix (e.g. the 4 × 4 upper triangular matrix (a_ij) with a₁₁ = a₂₂ = -3, a₃₃ = 1,a₄₄ = 0 has the multiset {-3, - 3, 1, 0}) and roots of a polynomial can be considered as multisets.

Zadeh [37] 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]. Yager [36] introduced the concept of Fuzzy Multiset and investigated a calculus for them. An element of a Fuzzy Multiset can occur more than once with possibly the same or different membership values.

In [30], Shinoj introduced fuzzy multi-groups and studied their properties. And in [25], Onasanya and Hoskova-Mayerova introduced multi-fuzzy groups induced by multisets. Our paper generalizes the work in [25 , 30] to polygroups. More specifically, it is concerned about (fuzzy) multi-polygroups and it is constructed as follows: After an Introduction, Section 2 presents basic notions related to polygroups and (fuzzy) multisets that are used throughout the paper. Section 3 defines and studies the properties of fuzzy multi-polygroups. Finally, Section 4 defines multi-polygroups, studies their properties and constructs fuzzy multi-polygroups from multi-polygroups.

2 Preliminaries

In this section, we present some definitions and theorems related to polygroups and (fuzzy) multisets that are used throughout the paper.

2.1 Polygroups

Let H be a non-empty set. Then, a mapping $\circ : H \times H \to P^{*} (H)$ is called a binary hyperoperation on H, where $P^{*} (H)$ is the family of all non-empty subsets of H. The couple (H, ∘) is called a hypergroupoid. See e.g. [4 , 23].

In the above definition, if A and B are two non-empty subsets of P and x ∈ H, then we define: $A \circ B = ⋃_{\binom{a \in A}{b \in B}} a \circ b, x \circ A = {x} \circ A$ and A ∘ x = A ∘ {x}.

A polygroup is a completely regular, reversible in itself multigroup in the sense of Dresher and Ore [16]. These systems occur naturally in the study of algebraic logic. Some algebraic and combinatorial properties were developed in [9].

Definition 2.1. [7, 14] A polygroup is a system <P,·,e,^-1>, where e ∈ P, ^-1 is an identity element, “·” maps P × P into the non-empty subsets of P, and the following axioms hold for all x, y, z ∈ P:

(x · y) · z = x · (y · z),

e · x = x · e = x,

x ∈ y · z implies y ∈ x · z^-1 and z ∈ y^-1 · x.

The following elementary facts about polygroups follow easily from the axioms: e ∈ x · x^-1 ∩ x^-1 · x, e^-1 = e, (x^-1) ^-1 = x, and (x · y) ^-1 = y^-1 · x^-1, where A^-1 = {a^-1| a ∈ A}.

Example 1. Let P₁ = {e, a} and define the commutative polygroup <P₁, ∘ ₁, e, ^-1 > by the following table: $\begin{matrix} \circ_{1} & e & a \\ e & e & a \\ a & a & {e, a} \end{matrix}$

Example 2. Let P₂ = {e, b, c} and define the commutative polygroup <P₂, ∘ ₂, e, ^-1 > by the following table: $\begin{matrix} \circ_{2} & e & b & c \\ e & e & b & c \\ b & b & {e, c} & {b, c} \\ c & c & {b, c} & {e, b} \end{matrix}$ Let <P, ∘ , e, ^-1 > be a polygroup and K ⊆ P. Then <K, ∘ , e, ^-1 > is a subpolygroup of (P, ∘ , e, ^-1) if for all a, b ∈ K, we have that a ∘ b ⊆ K and a^-1 ∈ K.

Definition 2.2. [14] Let systems <P₁, ∘ ₁, e₁, ^-1 > and <P_2,°_2,e_2,^-1> be polygroups with identities e₁ and e₂, respectively and f : P₁ → P₂ be a mapping with f (e₁) = e₂. Then f is:

a homomorphism if f (x ∘ ₁y) ⊆ f (x) ∘ ₂f (y) for all x, y ∈ P₁;

a strong homomorphism if f (x ∘ ₁y) = f (x) ∘ ₂f (y) for all x, y ∈ P₁;

an isomorphism if f is a bijective strong homomorphism.

Definition 2.3. [14] Let systems <M, ∘ ₁, e₁, ^-1 > , < N, ∘ ₂, e₂, ^-1 > be two polygroups. Then the productional polygroup <M× N, ∘ , (e₁, e₂) , ^-1 > is defined as (x₁, y₁) ∘ (x₂, y₂) = {(x₃, y₃) : x₃ ∈ x₁ ∘ ₁x₂, y₃ ∈ y₁ ∘ ₂y₂}. In the above definition, (M × N, ∘) with identity (e₁, e₂) and unitary operation ^-1 is a polygroup.

Example 3. Let <P₁, ∘ ₁, e, ^-1 > be the polygroup defined in Example 1. Then the productional polygroup <P₁× P₁, ∘ , (e, e) , ^-1 > is given by the following table. $\begin{matrix} \circ & (e, e) & (e, a) & (a, e) & (a, a) \\ (e, e) & (e, e) & (e, a) & (a, e) & (a, a) \\ (e, a) & (e, a) & {(e, a), (e,, e)} & (a, a) & {(a, a), (a, e)} \\ (a, e) & (a, e) & (a, a) & {(a, e), (e, e)} & {(e, a), (a, a)} \\ (a, a) & (a, a) & {(a, a), (a, e)} & {(e, a), (a, a)} & {(e, e), (e, a), (e, a), (a, a)} \end{matrix}$

Definition 2.4. [8] Let <M, ∘ ₁, e, ^-1 > , < N, ∘₂, e, ^-1> be two polygroups whose elements are rearranged so that M ∩ N = {e}. A new system M [N] =< P, ★ , e, ^I > called the extension of M by N is formed in the following way: Set P = M ∪ N and let e^I = e, x^I = x^-1, x ★ e = e ★ x = x for all x ∈ P, and for all x, y ∈ P \ {e} $x ★ y = {\begin{matrix} x \circ_{1} y, & if x, y \in M; \\ x, & if x \in N, y \in M; \\ y, & if x \in M, y \in N; \\ x \circ_{2} y, & if x, y \in N and x^{- 1} \neq y; \\ x \circ_{2} y \cup A, & if x, y \in N and x^{- 1} = y . \end{matrix}$

Theorem 2.5. [8] Let <M, ∘ ₁, e, ^-1 > , < N, ∘ ₂, e, ^-1> be two polygroups whose elements are rearranged so that M ∩ N = {e}. Then M [N] =< P, ★ , e, ^I >, the extension of M by N, is a polygroup.

Example 4. Let <P₁, ∘ ₁, e, ^-1 > and <P₂, ∘ ₂, e, ^-1> be the polygroups in Examples 1 and 2. Then the extension polygroup P₁ by P₂, (P₁ [P₂] , ★) is given by the following table. $\begin{matrix} ★ & e & a & b & c \\ e & e & a & b & c \\ a & a & {e, a} & b & c \\ b & b & b & {e, a, c} & {b, c} \\ c & c & c & {b, c} & {e, a, b} \end{matrix}$

More recent results concerning the polygroups can be found in [1 , 28].

2.2 Fuzzy multisets

Let us recall that the multiset is a collection of objects that can be repeated. A (crisp) multiset M is characterized by a count function C_M : X → N.

Fuzzy set and crisp set are the part of the distinct set theories, where the fuzzy set implements infinite-valued logic while crisp set employs bi-valued logic. Previously, expert system principles were formulated premised on Boolean logic where crisp sets are used. But then scientists argued that human thinking does not always follow crisp “yes”/“no” logic, and it could be vague, qualitative, uncertain, imprecise or fuzzy in nature. This gave commencement to the development of the fuzzy set theory to imitate human thinking.

Definition 2.6. [19] Let X be a set. A multiset M drawn from X is represented by a function C_M : X → {0, 1, 2, …}. For each x ∈ X, C_M (x) denotes the number of occurrences of x in M.

Example 5. Let X = {a, b, c, d}. Then M = {a, a, b, b, b, d, d, d, d, d} is a multiset. Here, C_M (a) =2, C_M (b) =3, C_M (c) =0 and C_M (d) =5.

Definition 2.7. [19] Let X be a set and M, N be multisets drawn from X. Then

M ⊆ N if C_M (x) ≤ C_N (x) for all x ∈ P,

The multiset M ∩ N is defined by C_M∩N where C_M∩N (x) = C_M (x) ∧ C_N (x) = min {C_M (x) , C_N (x)}.

Definition 2.8. [29] Let X be a non-empty set. A fuzzy multiset A drawn from X is represented by a function CM_A : X → Q where Q is the set of all crisp multisets drawn from the unit interval [0, 1].

In the above definition, the value CM_A (x) is a crisp multiset drawn from [0, 1]. For each x ∈ X, the membership sequence is defined as the decreasingly ordered sequence of elements in CM_A (x) and it is denoted by: $\begin{array}{l} {μ_{A}^{1} (x), μ_{A}^{2} (x), \dots, μ_{A}^{p} (x)} : \\ μ_{A}^{1} (x) \geq μ_{A}^{2} (x) \geq \dots \geq μ_{A}^{p} (x) . \end{array}$

Example 6. Let X = {a, b, c, d}. Then A = {(0.5, 0.5, 0.3)/a, (0.7, 0.2, 0.1, 0.1)/d} is a fuzzy multiset.

Let A, B be fuzzy multisets of X. Then L (x ; A) = $max {j : μ_{A}^{j} (x) \neq 0}$ and L (x) = L (x ; A, B) = max {L (x ; A) , L (x ; B)}. When we define an operation between two fuzzy multisets, the length of their membership sequences should be set equal.

Definition 2.9. [32] Let A, B be fuzzy multisets of X. Then

A ⊆ B if and only if $μ_{A}^{j} (x) \leq μ_{B}^{j} (x)$ , j = 1, …, L (x) for all x ∈ X,

A = B if and only if $μ_{A}^{j} (x) = μ_{B}^{j} (x)$ , j = 1, …, L (x) for all x ∈ X,

A ∩ B is defined by $μ_{A \cap B}^{j} (x) = μ_{A}^{j} (x) \land μ_{B}^{j} (x)$ , j = 1, …, L (x) for all x ∈ X.

Definition 2.10. [30] Let X, Y be non-empty sets, f : X → Y be a mapping, and A a fuzzy multiset of X and B a fuzzy multiset of Y. Then

The image of A under f is denoted by f (A) or ${CM}_{f (A)} (y) = {\begin{matrix} ⋁_{f (x) = y} {CM}_{A} (x) if f^{- 1} (y) \neq \emptyset \\ 0 otherwise . \end{matrix}$

The inverse image of B under f is denoted by f^-1 (B) where CM_f^-1(B) (x) = CM_B (f (x)).

3 Properties of fuzzy multi-polygroups

Inspired by the definition of fuzzy multigroups [30], we introduce the concept of fuzzy multi-polygroup. And we investigate their properties. It is well known that a polygroup is a generalized case of a group (i.e. every group is a polygroup and the converse may not hold.). So, the results in this section can be considered as more general than that in [30].

Definition 3.1. Let <P, ∘ , e, ^-1 > be a polygroup. A fuzzy multiset A over P is a fuzzy multi-polygroup of P if for all x, y ∈ P, the following conditions hold.

CM_A (x) ∧ CM_A (y) ≤ inf {CM_A (z) : z ∈ x ∘ y};

CM_A (x) ≤ CM_A (x^-1).

Example 7. Let <P₁, ∘ ₁, e, ^-1 > and <P₂, ∘ ₂, e, ^-1> be the polygroups defined in Examples 1 and 2. Then A = {(0.9, 0.8, 0.8, 0.7)/e, (0.7, 0 . 6, 0.6)/a} is a fuzzy multi-polygroup of P₁ and B = {(0.8, 0.8, 0.6, 0.5, 0.4)/e, (0.7, 0.3, 0.1, 0.1)/ b, (0.7, 0.3, 0.1, 0.1)/c} is a fuzzy multi-polygroup of P₂.

Proposition 3.2. Let <P, ∘ , e, ^-1 > be a polygroup with identity element e and A be a fuzzy multi-polygroup of P. Then the following are true.

CM_A (x) ≤ CM_A (e) for all x ∈ P;

CM_A (x^-1) = CM_A (x) for all x ∈ P;

CM_A (z) ≥ CM_A (x₁) ∧ … ∧ CM_A (x_n) for all z ∈ x₁ ∘ … ∘ x_n and n ≥ 2;

CM_A (z) ≥ CM_A (x) for all z ∈ xⁿ.

Proof.

Proof of 1.

Let x ∈ P. Then e ∈ x ∘ x^-1. The latter implies that CM_A (x) ≤ CM_A (x) ∧ CM_A (x^-1) ≤ C M_A (e).

Proof of 2.

For all x ∈ P, we have that x^-1 ∈ P, CM_A (x^-1) ≥ CM_A (x) and CM_A (x) = CM_A ( (x^-1) ^-1) ≥ CM_A (x^-1). Thus, CM_A (x^-1) = CM_A (x).

Proof of 3.

By mathematical induction on the value of n, CM_A (z) ≥ CM_A (x₁) ∧ … ∧ CM_A (x_n) for all z ∈ x₁ ∘ … ∘ x_n is true for n = 2. Assume that CM_A (z) ≥ CM_A (x₁) ∧ … ∧ CM_A (x_n) for all z ∈ x₁ ∘ … ∘ x_n and let z′ ∈ x₁ ∘ … ∘ x_n ∘ x_n+1. Then there exists x ∈ x₁ ∘ … ∘ x_n such that z′ ∈ x ∘ x_n+1. Having A a fuzzy multi-polygroup implies that CM_A (z′) ≥ CM_A (x) ∧ CM_A (x_n+1). And using our assumption that CM_A (x) ≥ CM_A (x₁) ∧ … ∧ CM_A (x_n) implies that our statement is true for n + 1.

Proof of 4.

The proof follows from 3. by setting x_i = x for all i = 1, …, n.□

Example 8. Let <P, ∘ , e, ^-1 > be any polygroup with identity e and A be a fuzzy multiset of P defined as CM_A (x) = CM_A (e) for all x ∈ P. Then A is a fuzzy multi-polygroup of P.

Remark 1. Let <P, ∘ , e, ^-1 > be a polygroup. Then we can define at least one fuzzy multi-polygroup of P which is mainly the one that is defined in Example 8.

Proposition 3.3. 1s Let <P, ∘ , e, ^-1 > be a polygroup and A be a fuzzy multiset of P. Then A is a fuzzy multi-polygroup of P if and only if CM_A (z) ≥ CM_A (x) ∧ CM_A (y) for all z ∈ x ∘ y^-1.

Proof. Let A be a fuzzy multi-polygroup of P and z ∈ x ∘ y^-1. Then CM_A (z) ≥ CM_A (x) ∧ CM_A (y^-1). Having CM_A (y^-1) ≥ CM_A (y) implies that CM_A (z) ≥CM_A (x) wedgeCM_A (y). Conversely, let A be a fuzzy multiset of P. Since e ∈ x ∘ x^-1, it follows that CM_A (e) ≥ CM_A (x) ∧ CM_A (x) = CM_A (x). The latter and having e ∘ x^-1 = x^-1 implies that CM_A (x^-1) ≥ CM_A (e) ∧ CM_A (x) = CM_A (x). Let z ∈ x ∘ y. Then z ∈ x ∘ (y^-1) ^-1. Thus, CM_A (z) ≥ CM_A (x) ∧ CM_A (y^-1) ≥ CM_A (x) ∧ CM_A (y).□

Definition 3.4. Let <P, ∘ , e, ^-1 > be a polygroup with identity element e and A be a fuzzy multiset of P. Then A^★ = {x ∈ P : CM_A (x) = CM_A (e)} and A_★ = {x ∈ P : CM_A (x) >0}.

Proposition 3.5. Let <P, ∘ , e, ^-1 > be a polygroup and A be a fuzzy multi-polygroup of P. Then A^★ is a subpolygroup of P.

Proof. We have A^★≠ ∅ as e ∈ A^★. Let x, y ∈ A^★. We need to show that x^-1 ∈ A^★ and x ∘ y ⊆ A^★. Having CM_A (x^-1) = CM_A (x) (by Proposition 3.2) implies that CM_A (x^-1) = CM_A (e) and hence, x^-1 ∈ A^★. Let z ∈ x ∘ y. Then CM_A (z) ≥ CM_A (x) ∧ CM_A (y) = CM_A (e). The latter and having CM_A (z) ≤ CM_A (e) (by Proposition 3.2) implies that CM_A (z) = CM_A (e) and hence, z ∈ A^★.□

Proposition 3.6. Let <P, ∘ , e, ^-1 > be a polygroup and A be a fuzzy multi-polygroup of P. Then A_★ is either an empty set or a subpolygroup of P.

Proof. The proof is straightforward.□

Definition 3.7. Let <P, ∘ , e, ^-1 > be a polygroup and A, B be fuzzy multisets of P. Then A ∘ B and A^-1 are defined by the following fuzzy count functions respectively. ${CM}_{A \circ B} (x) = \lor {{CM}_{A} (y) \land {CM}_{B} (z) : x \in y \circ z},$ ${CM}_{A^{- 1}} (x) = {CM}_{A} (x^{- 1}) .$

Remark 2. If A is a fuzzy multi-polygroup of P then A = A^-1.

Theorem 3.8. if Let <P, ∘ , e, ^-1 > be a polygroup and A be a fuzzy multiset of P. Then A is a fuzzy multi-polygroup of P if and only if A ∘ A ⊆ A and A^-1 = A.

Proof. Let A be a fuzzy multi-polygroup of P. Remark 2 asserts that A = A^-1. Let x, y ∈ P and z ∈ x ∘ y. Then CM_A (z) ≥ CM_A (x) ∧ CM_A (y). The latter implies that CM_A (z) ≥ ∨ {CM_A (x) ∧ CM_B (y) : z ∈ x ∘ y}. Thus, A ∘ A ⊆ A. Conversely, having A^-1 = A implies that CM_A (x) = CM_{A
^-1} (x) = CM_A (x^-1). And having A ∘ A ⊆ A implies that CM_A (z) ≥ CM_A∘A (z) = ∨ {CM_A (x) ∧ CM_A (y) : z ∈ x ∘ y}. The latter implies that CM_A (z) ≥ CM_A (x) ∧ CM_A (y) for all z ∈ x ∘ y. Thus, A is a fuzzy multi-polygroup of P.□

Remark 3. If A is a fuzzy multi-polygroup of P then A ⊆ A ∘ A. This is clear as x = x ∘ e and CM_A∘A (x) ≥ CM_A (x) ∧ CM_A (e) = CM_A (x).

Corollary 3.9. Let <P, ∘ , e, ^-1 > be a polygroup and A be a fuzzy multiset of P. Then A is a fuzzy multi-polygroup of P if and only if A ∘ A = A and A^-1 = A.

Proof. The proof follows from Theorem 3.8 and Remark 3.□

Proposition 3.10. Let <P, ∘ , e, ^-1 > be a polygroup and A, B be fuzzy multisets of P. If A and B are fuzzy multi-polygroups of P then A ∩ B is a fuzzy multi-polygroup of P.

Proof. Let x, y ∈ P and z ∈ x ∘ y. Then CM_A∩B (z) = CM_A (z) ∧ CM_B (z). Having A, B fuzzy multi-poly-groups of P implies that CM_A (z)≥ CM_A (x) ∧ CM_A (y) and CM_B (z) ≥ CM_B (x) ∧ C M_B (y). The latter implies that CM_A∩B (z) ≥ CM_A (x) ∧ CM_A (y) ∧ CM_B (x) ∧ CM_B (y) = CM_A∩B (x) ∧ CM_A∩B (y). Moreover, CM_A∩B (x^-1) = CM_A (x^-1) ∧ CM_B (x^-1) ≥ CM_A (x) ∧ CM_B (x) = CM_A∩B (x). Therefore, A ∩ B is a fuzzy multi-polygroup of P.□

Corollary 3.11. Let <P, ∘ , e, ^-1 > be a polygroup and A_i be fuzzy multiset of P for all i = 1, 2, …, n. If A_i is a fuzzy multi-polygroup of P then $⋂_{i = 1}^{n} A_{i}$ is a fuzzy multi-polygroup of P.

Proof. The proof results from using mathematical induction and Proposition 3.10.□

Example 9. Let <P₁, ∘ ₁, e, ^-1 > be the polygroup defined in Example 1 and A, B be the fuzzy multi-polygroups of P₁ defined by: $A = {(0.7, 0.6, 0.5) / e, (0.4, 0.3, 0.3) / a},$ $B = {(0.8, 0.3, 0.3, 0.1) / e, (0.5, 0.2, 0.1) / a} .$ Then A ∩ B = {(0.7, 0.3, 0.3)/e, (0.4, 0.2, 0.1)/a} is a fuzzy multi-polygroup of P₁.

Proposition 3.12. Let systems <P₁, ∘ ₁, e₁, ^-1 > and <P₂, ∘ ₂, e₂, ^-1 > be two polygroups with fuzzy multisets A, B respectively. If A and B are fuzzy multi-polygroups of P₁ and P₂ respectively then A × B is a fuzzy multi-polygroup of the productional polygroup <P₁× P₂, ∘ , (e₁, e₂) , ^-1 >, where for all (x, y) ∈ P₁ × P₂, CM_A×B (x, y) = CM_A (x) ∧ CM_B (y).

Proof. Let (x, y) ∈ P₁ × P₂. Then (x, y) ^-1 = (x^-1, y^-1). We have $\begin{array}{l} C M_{A \times B} ((x, y)^{-} 1) = C M_{A} \times B ((x^{-} 1, y^{-} 1)) \\ = C M_{A} (x^{-} 1) \land C M_{B} (y^{-} 1) . \end{array}$

Since A, B are fuzzy multi-polygroups of P₁, P₂ respectively, it follows that CM_A (x^-1) = CM_A (x) and CM_B (y^-1) = CM_B (y). Thus, CM_A×B ((x, y) ^-1) = CM_A (x) ∧ CM_B (y) = CM_A×B (x, y). Let (x₃, y₃) ∈ (x₁, y₁) ∘ (x₂, y₂). Then x₃ ∈ x₁ ∘ x₂ and y₃ ∈ y₁ ∘ y₂. Having A, B are fuzzy multi-polygroups of P₁, P₂ respectively implies that CM_A (x₃) ≥ CM_A (x₁) ∧ CM_A (x₂) and CM_B (y₃) ≥ CM_B (y₁) ∧ CM_B (y₂). We get now $C M_{A} \times B (x_{3}, y_{3}) = C M_{A} (x_{3}) \land C M_{B} (y_{3}) \geq C M_{A} (x_{1}) \land C M_{A} (x_{2}) \land C M_{B} (y_{1}) \land C M_{B} (y_{2}) .$ The latter implies that $\begin{matrix} {CM}_{A \times B} (x_{3}, y_{3}) \geq {CM}_{A \times B} (x_{1}, y_{1}) \land \\ {CM}_{A \times B} (x_{2}, y_{2}) . \end{matrix}$ □

Example 10. Let (P₁ × P₁, ∘) be the productional polygroup defined in Example 1 and let A = {(0.8, 0.3, 0.3, 0.1)/e, (0.5, 0.2, 0.1)/a} be the fuzzy multi-polygroup of P₁.

Then A × A = {(0.8, 0.3, 0.3, 0.1)/(e, e) , (0.5, 0.2, 0.1)/(e, a) , (0.5, 0.2, 0.1)/(a, e) , (0.5, 0.2, 0.1)/(a, a)} is a fuzzy multi-polygroup of P₁ × P₂.

Proposition 3.13. Let M and N be two polygroups such that M ∩ N = {e}. Define the fuzzy multi-polygroups CM_A, CM_B of M and N respectively. If CM_A (x) ≥ CM_B (y) for all (x, y) ∈ M × N then CM is a fuzzy multi-polygroup of M [N], where for all x ∈ M [N], $CM (x) = {\begin{matrix} {CM}_{A} (e), & if x = e; \\ {CM}_{A} (x), & if x \neq e \in M; \\ {CM}_{B} (x), & if x \neq e \in N . \end{matrix}$

Proof. We need to prove that the conditions of Definition 3.1 are satisfied for CM. If x = e or y = e then the conditions hold trivially. Let x, y ≠ e ∈ M [N]. We have the following cases for x, y:

Case x, y ∈ M. For all z ∈ x ★ y = x ∘ ₁y, we have CM (z) = CM_A (z) ≥ min {CM_A (x) , CM_A (y)} = min {CM (x) , CM (y)}

Case x, y ∈ N and y ≠ x^-1. For all z ∈ x ★ y = x ∘ ₂y, we have CM (z) = CM_B (z) ≥ min {CM_B (x) , CM_B (y)} = min {CM (x) , CM (y)}.

Case x, y ∈ B and y = x^-1.

For all z ∈ x ★ y = x ∘ ₂y ∪ A, we have

CM (z) = CM_B (z) ≥ min {CM_B (x) , CM_B (y)} = min {CM (x) , CM (y)} if z ∈ x ∘ ₂y. If z ∈ A, we have

CM (z) = CM_A (z) ≥ min {CM_B (x) , CM_B (y)} = min {CM (x) , CM (y)}.

Case x ∈ A and y ∈ B. For all z ∈ x ★ y = y, we have

CM (z) = CM_B (y) ≥ min {CM_A (x) , CM_B (y)} = min {CM (x) , CM (y)}.

Case x ∈ B and y ∈ A. For all z ∈ x ★ y = x, we have

CM (z) = CM_B (x) ≥ min {CM_B (x) , CM_A (y)} = min {CM (x) , CM (y)}.

For all x ≠ e ∈ M [N],

$CM (x^{I}) = {\begin{matrix} {CM}_{A} (x^{- 1}), & if x \in M; \\ {CM}_{B} (x^{- 1}), & if x \in N . \end{matrix} \geq CM (x)$ .

□

Example 11. Let A = {(0.9, 0.8, 0.8, 0.7)/e, (0.8, 0.7, 0.6)/a} be a fuzzy multi-polygroup of the polygroup P₁ defined in Example 1 and B = {(0.8, 0.7, 0.6) /e, (0.6, 0.5, 0.5)/b, (0.6, 0.5, 0.5)/c} be a fuzzy multi-polygroup of the polygroup P₂ defined in Example 2. Using Proposition 3.13, we get that C = {(0.9, 0.8, 0.8, 0.7)/e, (0.8, 0.7, 0.6)/a, (0.6, 0.5, 0.5) /b, (0.6, 0.5, 0.5)/c} is a fuzzy multi-polygroup of the polygroup P₁ [P₂] presented in Example 4.

Proposition 3.14. Let system <P₁, ∘ ₁, e, ^-1 > and system <P₂, ∘ ₂, e, ^-1 > be polygroups, A, B be fuzzy multi sets of P₁, P₂, respectively and f : P₁ → P₂ be a strong homomorphism. Then

If A is a fuzzy multi-polygroup of P₁, the f (A) is a fuzzy multi-polygroup of P₂,

If B is a fuzzy multi-polygroup of P₂, the f^-1 (B) is a fuzzy multi-polygroup of P₁.

Proof. Item 1.: Let y₁, y₂ ∈ P₂ and y₃ ∈ y₁ ∘ ₂y₂. If f^-1 (y₁) =∅ or f^-1 (y₂) =∅ then CM_f(A) (y₁) =0 or CM_f(A) (y₂) =0. We get that CM_f(A) (y₃) ≥0 = CM_f(A) (y₁) ∧ M_f(A) (y₂). If f^-1 (y₁)≠ ∅ and f^-1 (y₂)≠ ∅ then there exist x₁, x₂ ∈ P₁ such that CM_A (x₁) = ⋁ _f(x)=y₁CM_A (x) and CM_A (x₂) = ⋁ _f(x)=y₂CM_A (x). Having f a homomorphism implies that y₃ ∈ f (x₁) ∘ ₂f (x₂) = f (x₁ ∘ ₁x₂). The latter implies that there exists x₃ ∈ x₁ ∘ x₂ such that y₃ = f (x₃). Since A is a fuzzy multi-polygroup of P₁, it follows that CM_f(A) (y₃) ≥ CM_A (x₃) ≥ CM_A (x₁) ∧ CM_A (x₂) = CM_f(A) (y₁) ∧ CM_f(A) (y₂). We still need to prove that CM_f(A) (y^-1) ≥ CM_f(A) (y). We have

CM_f(A) (y^-1) =

$= {\begin{matrix} ⋁_{f (x) = y^{- 1}} {CM}_{A} (x) if f^{- 1} (y^{- 1}) \neq \emptyset \\ 0 otherwise . \end{matrix}$ $= {\begin{matrix} ⋁_{f (x^{- 1}) = y} {CM}_{A} (x^{- 1}) if f^{- 1} (y) \neq \emptyset \\ 0 otherwise . \end{matrix}$ = CM_f(A) (y).

Item 2.: Let x₁, x₂ ∈ P₁ and x₃ ∈ x₁ ∘ ₁x₂. Then CM_f^-1(B) (x₃) = CM_B (f (x₃)).

Having f (x₃) ∈ f (x₁ ∘ ₁x₂) = f (x₁) ∘ f (x₂) im-plies that CM_f^-1(B) (x₃) = CM_B (f (x₃)) ≥ CM_B (f (x₁)) ∧ CM_B (f (x₂)) = CM_f^-1(B) (x₁) ∧ CM_f^-1(B) (x₂). Let x ∈ P₁. Then CM_f^-1(B) (x^-1) = CM_B (f (x^-1)).

Having f (x^-1) = (f (x)) ^-1 and CM_B (f (x)) = CM_B (f (x)) ^-1) (as B is a fuzzy multi-polygroup of P₂) implies that CM_f^-1(B) (x^-1) = CM_f^-1(B) (x).□

4 Fuzzy multi-polygroups induced bymultisets

In this section, we define multi-polygroups of polygroups using multisets and we use our new definition to get fuzzy multi-polygroups.

Definition 4.1. A multiset M of a polygroup <P, ∘ , e, ^-1 > with a count function C_M is called a multi-polygroup of P if the following conditions hold.

C_M (x) ∧ C_M (y) ≤ inf {C_M (z) : z ∈ x ∘ y};

C_M (x) ≤ C_M (x^-1).

Example 12. Let system <P₁, ∘ ₁, e, ^-1 > and system <P₂, ∘ ₂, e, ^-1 > be the polygroups defined in Examples 1 and 2. Then A = {(e, e, e, e) , (a, a, a)} and B = {(e, e, e, e, e, e) , (b, b, b, b) , (c, c, c, c)} are multi-polygroups of <P₁, ∘ ₁, e, ^-1 > and <P₂, ∘ ₂, e, ^-1 > respectively.

Proposition 4.2. Let <P, ∘ , e, ^-1 > be a polygroup with identity element e and M be a multi-polygroup of P. Then the following are true.

C_M (x) ≤ C_M (e) for all x ∈ P;

C_M (x^-1) = C_M (x) for all x ∈ P;

C_M (z) ≥ C_M (x₁) ∧ … ∧ C_M (x_n) for all z ∈ x₁ ∘ … ∘ x_n;

C_M (z) ≥ C_M (x) for all z ∈ xⁿ.

Proof. The proof is similar to that of Proposition 3.2.

□

Proposition 4.3. Let <P, ∘ , e, ^-1 > be a polygroup and M, N be multisets of P. If M and N are multi-polygroups of P then M ∩ N is a multi-polygroup of P.

Proof. The proof is similar to that of Proposition 3.10.

□

Corollary 4.4. Let <P, ∘ , e, ^-1 > be a polygroup and M_i be a multiset of P for all i = 1, 2, …, n. If M_i is a multi-polygroup of P then $⋂_{i = 1}^{n} M_{i}$ is a multi-polygroup of P.

Proof. The proof results from the use of mathematical induction and Proposition 4.3.□

Example 13. Let <P₁, ∘ ₁, e, ^-1 > be the polygroup defined in Example 1 and A, B be the multi-polygroups of P₁ defined by: $M = {(e, e, e, e, e), (a, a)}, N = {(e, e, e, e), (a, a, a)} .$ Then M ∩ N = {(e, e, e, e) , (a, a)} is a multi-polygroup of P₁.

Proposition 4.5. Let system <P₁, ∘ ₁, e₁, ^-1 > and system <P₂, ∘ ₂, e₂, ^-1 > be two polygroups with multisets M, N, respectively. If A and B are multi-polygroups of P₁ and P₂ respectively then M × N is a multi-polygroup of the productional polygroup (P₁ × P₂, ∘), where for all (x, y) ∈ P₁ × P₂, C_M×N (x, y) = C_M (x) ∧ C_N (y).

Proof. The proof is similar to that of Proposition 3.12.

□

Proposition 4.6. Let P and Q be two polygroups such that P ∩ Q = {e}. Define the multi-polygroups C_M, C_N of P and Q respectively. If C_M (x) ≥ C_N (y) for all (x, y) ∈ P × Q then C_S is a multi-polygroup of P [Q], where for all x ∈ P [Q], $C_{S} (x) = {\begin{matrix} C_{M} (e), & if x = e; \\ C_{M} (x), & if x \neq e \in P; \\ C_{N} (x), & if x \neq e \in Q . \end{matrix}$

Proof. The proof is similar to that of Proposition 3.13.

□

Let M, N be multisets on a polygroup P and with C_M (x) ≤ α and C_N (x) = α for all x ∈ P. We define ${CM}_{\frac{M}{N}} (x) = \frac{C_{M (x)}}{α}$ . Having ${CM}_{\frac{M}{N}} (x) \leq 1$ for all x ∈ P implies that $M = \frac{M}{N}$ is a fuzzy multiset of P.

Proposition 4.7. Let M, N be multisets on a polygroup <P, ∘ , e, ^-1 > with C_M (x) ≤ α and C_N (x) = α for all x ∈ P. If M is a multi-polygroup of P then $M$ is a fuzzy multi-polygroup of P.

Proof. Let x, y ∈ P and z ∈ x ∘ y^-1. ${CM}_{M} (z) = \frac{C_{M (z)}}{α}$ . Since C_M (z) ≥ C_M (x) ∧ C_M (y^-1) = C_M (x) ∧ C_M (y), it follows that ${CM}_{M} (z) \geq {CM}_{M} (x)$ $\land {CM}_{M} (y)$ . Applying Proposition 3.3 on the latter implies that $M$ is a fuzzy multi-polygroup of P.□

Example 14. Let M = {(e, e, e, e) , (b, b, b) , (c, c, c)} , N = {(e, e, e, e) , (b, b, b, b) , (c, c, c, c)} be multi-polygroups of the polygroup defined in Example 2. Then $M = {(1, 1, 1, 1) / e, (0.75, 0.75, 0.75) / b, (0.75, 0.75, 0.75) / c}$ is a fuzzy multi-polygroup of P.

5 Conclusion

This paper has introduced an algebraic hyperstructure of multisets and fuzzy multisets in the form of multi-polygroups and fuzzy multi-polygroups respectively. Several interesting properties of the new defined notions were discussed. It is well known that the concept of (fuzzy) multiset is well established in dealing with many real life problems. So, the algebraic hyperstructure defined on them in this paper would help to approach these problems with a different perspective.

Footnotes

Acknowledgment

The work presented in this paper was supported within the project for development of basic and applied research developed in the long term by the departments of theoretical and applied bases FMT (Project code: DZRO K-217) supported by the Ministry of Defence in the Czech Republic.

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