(α, γ)-Anti-Multi-Fuzzy Subgroups and Some of Its Properties

1  Introduction

Dresher et al. [1] laid the foundations of the theory of multigroup in 1938. Zadeh [2] introduced the concept of a fuzzy subset of a set, fuzzy set are a kind of useful mathematical structure to represent a collection of objects whose boundary is uncertainty in 1965. Therefore, on the basis of fuzzy set theory, Sebastian et al. [3] introduced Multi-Fuzzy Sets, Atanassov [4] proposed intuitionistic fuzzy set theory, Shinoj et al. [5] initiated intuitionistic fuzzy multisets. Recently, the above theories have developed in many directions and found its applications in a wide variety of fields including algebraic structures. For example, on fuzzy sets [6–8], on fuzzy multi sets [9–11] on anti-fuzzy group theory [12–17] are some of the selected works. Rosenfeld [18] defined the notion of fuzzy subgroup. Biswas [19] introduced the concept of anti-fuzzy subgroup of group. Yuan et al. [20] introduced the concept of fuzzy subgroup with thresholds. A fuzzy subgroup with thresholds lambda and mu is also called a (lambda, mu)-fuzzy subgroup. Yao [21] defined (lambda, mu)-fuzzy normal subgroups and (lambda, mu)-fuzzy quotient subgroups these examined some properties. On these studies, Shen [22] defined anti-fuzzy subgroups and Dong [23] introduced the product of anti-fuzzy subgroups. Then, Feng et al. [24] introduces the notion of (lambda, mu)-anti-fuzzy subgroups and discussed some properties. Since the idea of anti-multi fuzzy subgroup has been extended to multi fuzzy subgroups, it is expedient to explore the idea in (α, γ)-anti-multi-fuzzy subgroups setting. The motivation of this paper is to extend the notions of anti-multi fuzzy subgroups and (α, γ)-anti-multi-fuzzy subgroups to fuzzy multigroup environment and to present some new results. Moreover, this research proposes the generalization of the results known for (α, γ)-anti-multi-fuzzy subgroups. It is known that the notion of fuzzy multiset is well entrenched in solving many real-life problems. So, the algebraic structure defined concerning them in this paper could help to approach these issues from a different position. The benefit of this paper is the link found between algebraic structures and fuzzy multisets by introducing (α, γ)-anti-multi-fuzzy subgroups and studying their properties.

The outlines are presented as follows: Section 2 presents some foundational notions relevant to the study, whereas the main results are reported in Section 3. In Section 4, we make some concluding remarks and suggestions for future work.

2  Preliminary

In this paper, G,G1 and G2 stands for groups with identities 1,11 and 12, respectively. In the rest of the article, we will always suppose that 0≤α<γ≤1.

Definition 2.1 [3] Let A be a fuzzy subset of G. A is called a fuzzy subgroup of G if, for all x.y∈G,

     i)  A(xy)≥A(x)∧A(y),

    ii)  A(x−1)≥A(x).

Definition 2.2 [9] Let A be a fuzzy subset of G. A is called a (α,γ)-anti-fuzzy subgroup of G if, for all x.y,z∈G,

     i)  A(xy)∧γ≤(A(x)∨A(y))∨α,

    ii)  A(z−1)∧α≤A(z)∨γ.

Definition 2.3 [10] Let E 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 E is represented by a function CMA:E→Q.

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

(μA1(x),μA2(x),…,μAp(x)):μA1(x)≥μA2(x)≥…≥μAp(x).

A fuzzy set on a set E can be understood as a special case of fuzzy multiset where CMA(x)=μA1(x) for all x∈E.

3  (α, γ)-Anti-Multi Fuzzy Subgroups and Some of Its Properties

Definition 3.1 A fuzzy set A of a group G is called a (α,γ)-anti fuzzy multi subgroup of G if ∀g1,g2,g3∈G

μGi(g1g2)∧γ≤(μGi(g1)∨μGi(g2))∨α(1)

and

μGi((g3)−1)∧γ≤μGi(g3)∨α(2)

where (g3)−1 is the inverse element of (g3).

Proposition 3.2 If A is a (α,γ)-anti-fuzzy-multi-subgroup of a group G, then

μGi(1)∧γ≤μGi(g1)∨α(3)

∀g1∈G, where 1 is the identity of G.

Proof ∀g1∈G and let (g1)−1 be the inverse element of (g1). Then

μGi(1)∧γ=μGi((g1)−1g1)∧γ≤(μGi((g1)−1g1)∧γ)∧γ≤((μGi(g1)∨μGi(g1)−1)∨α)∧γ=(μGi(g1)∧γ)∨(μGi(g1−1)∧γ)∨(α∧γ)≤μGi(g1)∨(μGi(g1)∨α)∨α=μGi(g1)∨α(4)

Theorem 3.3 Let A be multi fuzzy subset of a group G. Then A is a (α,γ)-anti-fuzzy multi subgroup of

G⇔μGi((g1)−1g2)∧γ≤(μGi(g1)∨μGi(g2))∨α,∀g1,g2∈G.(5)

Proof Let A is a (α,γ)-anti-fuzzy multi group of G, then

μGi((g1)−1g2)∧γ=μGi((g1)−1g2)∧γ∧γ≤((μGi(g2)∨μGi(g1)−1)∨α)∧γ=(μGi(g2)∨μGi(g1−1)∧γ)∨(α∧γ)≤μGi(g2)∨(μGi(g1)∨α)∨α=(μGi(g1)∨μGi(g2))∨α.(6)

Conversely, assume

μGi((g1)−1g2)∧γ≤(μGi(g1)∨μGi(g2))∨α,∀g1,g2∈G,(7)

then

μGi(1)∧γ=μGi((g1)−1g2)∧γ≤(μGi(g1)∨μGi(g1))∨α=μGi(g1)∨α.(8)

So

μGi((g1)−1)∧γ=μGi((g1)−11)∧γ=μGi((g1)−11)∧γ∧γ=(μGi(g1)∨μGi(1)∨α)∧γ≤μGi(g2)∨(μGi(g1)∨α)∨α=(μGi(g1)∨μGi(g2))∨α.(9)

In this way A is a (α,γ)-anti-fuzzy multi-subgroup of G.

Theorem 3.4 Let A be a fuzzy multi subset of a group G. Then the following are equivalent:

     i)  A(δ) is a multi-subgroup of G, ∀δ∈(α,γ], where A(δ)≠∅;

    ii)  A is a (α,γ)-anti-fuzzy multi subgroup of G.

Proof (i)⇒(ii) let A(δ) is a multi-subgroup of G. We need to prove that

μGi((g1)−1g2)∧γ≤μGi(g1)∨μGi(g2)∨α,∀g1,g2∈G.(10)

If there exists ∀g3,g4∈G such that

μGi((g3)−1g4)∧γ=δ>μGi(g3)∨μGi(g4)∨α,(11)

Then μGi(g3)<δ,μGi(g4)<δ and δ∈(α,γ]. Thus μGi(g3)∈A(δ),μGi(g4)∈A(δ). But μGi((g3)−1g4)≥δ, that is (g3)−1g4∉A(δ). This is a contradiction with that A(δ) is a multi-subgroup of G. Hence

μGi((g1)−1g2)∧γ≤μGi(g1)∨μGi(g2)∨α,(12)

Holds ∀g1,g2∈G. Therefore, A is a (α,γ)-anti-fuzzy multi subgroup of G.

(ii)⇒(i)

Let A is a (α,γ)-anti-fuzzy multi subgroup of G. ∀δ∈(α,γ], such that A(δ)≠∅, we need to show that (g1)−1g2∉A(δ), ∀g1,g2∈A(δ). Since μGi(g1)<δ,μGi(g2)<δ then

μGi((g1)−1g2)∧γ≤μGi(g1)∨μGi(g2)∨α<δ∨δ∨α=δ∨α=α.(13)

Note that δ<γ, we have μGi((g1)−1g2)<δ. Thus (g1)−1g2∈A(δ). We set inf∅=1, where ∅ is the empty set.

Theorem 3.5 Let A and B are two fuzzy multi-subsets of groups G1 and G2, respectively. The product of A and B, denoted by μG1i×μG2is a (α,γ)-anti-fuzzy multi-subgroup of G1×G2, where

μG1i×μG2i(g1,g2)=μG1i(g1)∨μG2i(g2),∀(g1,g2)∈G1×G2.(14)

Proof Let g21−1 is the inverse element of g21 in G2 and g11−1 is the inverse element of g11 in G1. Then(g11−1,g21−1) be the inverse element of (g11,g21)∈G1×G2. Hence

μG1i(g11−1)∧γ≤μG1i(g11)∨α(15)

and

μG2i(g21−1)∧γ≤μG2i(g21)∨α.(16)

∀(g12,g22)∈G1×G2. We have

((μG1i×μG2i)(g11,g21)−1,(g12,g22))∧γ=(μG1i×μG2i)((g11−1,g21−1),(g12,g22))∧γ=(μG1i(g11−1,g12)∨μG2i(g21−1,g22))∧γ=(μG1i(g11−1,g12)∧γ)∨(μG2i(g21−1,g22)∧γ)≤(μG1i(g11)∨μG1i(g12)∨α)∨(μG2i(g21)∨μG2i(g22)∨α)≤(μG1i(g11)∨μG1i(g12)∨α)∨(μG2i(g21)∨μG2i(g22)∨α)=(μG1i(g11)∨μG2i(g21))∨(μG1i(g12)∨μG2i(g22))∨α=(μG1i×μG2i(g11,g21))∨(μG1i×μG2i(g12,g22))∨α.(17)

Hence μG1i×μG2i s a (α,γ)-anti-fuzzy multi-subgroup of G1×G2.

Theorem 3.6 Let A and B are two fuzzy multi-subsets of groups G1 and G2, respectively. If A×B is a (α,γ)-anti-fuzzy multi-subgroup of G1×G2, then at least one of the following statements must hold.

μG1i(11)∧γ≤μG2i(a)∨α,∀a∈G2(18)

and

μG2i(12)∧γ≤μG1i(g1)∨α,∀g1∈G1.(19)

Proof Let A×B is a (α,γ)-anti-fuzzy multi-subgroup of G1×G2. By contraposition, assume that none of the statements holds. Then we can find g1∈G1 and a∈G2 such that μG1i(g1)∨α<μG2i(12)∧γ and μG2i(a)∨α<μG1i(11)∧γ. Now

μG1i×μG2i(g1,a)∨α=(μG1i(g1)∨μG2i(a))∨α,=(μG1i(g1)∨α)∨(μG2i(a)∨α)<(μG1i(11)∧γ)∨(μG2i(12)∧γ)=μG1i×μG2i(11,12)∧γ.(20)

Therefore A×B is a (α,γ)-anti-fuzzy multi-subgroup of G1×G2 satisfying

μG1i×μG2i(g1,a)∨α<μG1i×μG2i(11,12)∧γ.(21)

This is a contradict with that (11,12) is the identity of G1×G2.

Theorem 3.7 Let f:G1→G2 is a homomorphism and let A is a (α,γ)-anti-fuzzy multi-subgroup of G1. Then f(μG1i) is a (α,γ)-anti-fuzzy multi-subgroup of G2, where

f(μG1i)(g2)=infg1∈G1{μG1i(g1):f(g1)=g2},∀g2∈G2(22)

Proof If f−1(g21)=∅ or f−1(g22)=∅ for any g21,g22∈G2, then

(f(μG1i)((g21)−1g22))∧γ≤1=(f(μG1i)(g21)∨f(μG1i)(g22))∨α.(23)

Assume that f−1(g21)=∅ or f−1(g22)=∅ for any g21,g22∈G2 then

(f(μG1i)((g21)−1g22))∧γ=infk∈G1{μG1i(k):f(k)=(g21)−1g22}∧γ=infk∈G1{μG1i(k)∧γ:f(k)=(g21)−1g22}≤infg11,g12∈G1{μG1i((g11)−1g12)∧γ:f(g11)=g21,f(g12)=g22}≤infg11,g12∈G1{(μG1i)(g11)∨f(μG1i)(g12)∨α:f(g11)=g21,f(g12)=g22}=(infg11∈G1{(μG1i)(g11):f(g11)=g21}∨infg12∈G1{(μG1i)(g12):f(g12)=g22})∨α=(f(μG1i)(g21)∨f(μG1i)(g22))∨α(24)

Thus f(μG1i) is a (α,γ)-anti-fuzzy multi-subgroup of G2.

Theorem 3.8 Let f:G1→G2 is a homomorphism and let A is a (α,γ)-anti-fuzzy multi-subgroup of G2. Then f−1(μG2i) is a (α,γ)-anti-fuzzy multi-subgroup of G1, where

f−1(μG2i)(g1)=μG2i(f(g1)),∀g1∈G1.(25)

Proof For any g11,g12∈G1

f−1(μG2i)((g11)−1g12)∧γ=μG2i(f((g11)−1g12))∧γ=μG2i(f((g11)−1)f(g12))∧γ≤(μG2i(f(g11))∨μG2i(f(g12)))∨α=(f−1(μG2i)(g11)∨f−1(μG2i)(g12))∨α.

Thus f−1(μG2i) is a (α,γ)-anti-fuzzy multi-subgroup of G1.

Definition 3.9 Let A is a (α,γ)-anti fuzzy multi subgroup of G. A is called a (α,γ)-anti fuzzy normal subgroup of G if, ∀g1,g2∈G

μGi(g1g2g1−1)∧γ≤μGi(g2)∨α(26)

Proposition 3.10 Let A is a (α,γ)-anti fuzzy multi subgroup of G. A is a (α,γ)-anti fuzzy normal subgroup if and only if, ∀g1,g2∈G,

μGi(g1g2)∧γ≤μGi(g2g1)∨α.(27)

Proposition 3.11 Let A is a (α,γ)-anti fuzzy multi subset of G. Then A is a (α,γ)-anti fuzzy multi normal subgroup of G if and only if Aδ≠∅ is a normal subgroup of G ∀δ∈(α,γ].

Proposition 3.12 Let A is a (α,γ)-anti fuzzy multi normal subgroup of G and g1∈G.

     i)  If μGi(g1)≤α, then μGi(g1g2g1−1)≤α for all g2∈G.

    ii)  If γ<μGi(g1)<α, then μGi(g2g1g2−1)=μGi(g1) for all g2∈G.

   iii)  If g2∈G and γ<μGi(g1g2)<α, then μGi(g1g2)=μGi(g2g1).

    iv)  If g2∈G and μGi(g1g2)≤α, then, μGi(g2g1)≤α

     v)  If g2∈G and μGi(g1g2)≥γ, then, μGi(g2g1)≥γ.

Proof (i) μGi(g1)≤α, then g1∈Aα. By proposition 9, Aα is a fuzzy multi normal subgroup of G and thus g1g2g1−1∈Aα. Hence μGi(g1g2g1−1)≤α. (ii) Let μGi(g1)=δ. Then γ<δ<α. By proposition 9, Aδ is a fuzzy multi normal subgroup of G. Hence g1g2g1−1∈Aδ; that is μGi(g1g2g1−1)≤δ=μGi(g1). Suppose μGi(g1g2g1−1)<δ. Set δ0=min{μGi(g1g2g1−1),α}. Then γ<δ0<α. By proposition 9, Aδ0 is a fuzzy multi normal subgroup of G, and thus g1g2g1−1∈Aδ0. Therefore g1=g2−1(g1g2g1−1)g2∈Aδ0; that is μGi(g1g2g1−1)≤δ0<δ which is a contradiction to that μGi(g1)=δ. As result, μGi(g2g1g2−1)=μGi(g1).

(iii) If γ<μGi(g1g2)<α, then μGi(g2g1)=μGi(g1−1(g2g1)g1)=μGi(g1g2) by (ii); that is μGi(g1g2)=μGi(g2g1).

(iv) μGi(g1g2)≤α, then g1g2∈Aα. Since Aα is a fuzzy multi normal subgroup of G by Proposition 9, (g2g1)=g1−1(g1g2)g1∈Aα; that is μGi(g2g1)≤α.

(v) Assume that μGi(g2g1)<γ on the contrary. If μGi(g2g1)≥α, then, by (i) μGi(g1g2)≥α, which is contradictory to that μGi(g1g2)>γ. If μGi(g2g1)<γ, then, by (iii), μGi(g1g2)=μGi(g2g1)<α, which is contradictory to that μGi(g1g2)≥γ. Therefore μGi(g2g1)≥γ.

Proposition 3.13 Let A is a (α,γ)-anti fuzzy multi subgroup of G. Then A is a (α,γ)-anti fuzzy multi normal subgroup of G if and only if

μGi([g1,g2])∧γ≤μGi(g1)∨α(28)

for all g1,g2∈G, where [g1,g2]=g1−1g2−1g1g2 is a commutator in G.

Proof For any g1,g2∈G,

μGi([g1,g2])∧γ=μGi(g1−1g2−1g1g2)∧γ=μGi(g1−1(g2−1g1g2))∧γ∧γ≤(μGi(g1−1)∨μGi(g2−1g1g2)∨α)∧γ=(μGi(g1−1)∧γ)∨(μGi(g2−1g1g2)∧γ)∨α(29)

Since A is a (α,γ)-anti fuzzy multi normal subgroup of G, μGi(g1−1)∧γ≤μGi(g1)∨α and μGi(g2−1g1g2)∧γ≤μGi(g1)∨α. Hence,

μGi([g1,g2])∧γ≤μGi(g1)∨α(30)

Conversely, if μGi([g1,g2])∧γ≤μGi(g1)∨α, then

μGi([g1,g2])∧γ=μGi(g1g1−1g2−1g1g2)∧γ=μGi(g1(g1−1g2−1g1g2))∧γ∧γ≤(μGi(g1)∨μGi(g1−1g2−1g1g2)∨α)∧γ=(μGi(g1)∧γ)∨(μGi(g1−1g2−1g1g2)∧γ)∨α≤(μGi(g1)∧γ)∨(μGi(g1)∨α)∨α≤μGi(g1)∨μGi(g1)∨α=μGi(g1)∨α(31)