# Belligerent Fuzzy GE-Filters on GE-Algebras

Authors: Teferi Getachew Alemayehu; Yeshiwas Mebrat Gubena
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, the concept of belligerent fuzzy GE-filter of GE-algebra is introduced. The relationship between a fuzzy GE-filter of GE-algebra and a belligerent fuzzy GE-filter of GE-algebra is given. Further, it is shown that a finite product (union) of belligerent fuzzy GE-filters of GE-algebras is a belligerent fuzzy GE-filter of the finite product (union) of GE-algebras.


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## Body

## 1. Introduction

In 1966, Diego [1] introduced the concept of Hilbert algebras. Diego discussed several properties of Hilbert algebras and deductive systems. More studies on the ideas of Hilbert algebras and deductive systems were performed by Busneag [2, 3]. Bandaru discussed a new algebraic structure, called GE-algebra [4] as a generalization of Hilbert algebra. Filters, upper sets, and congruence kernels in GE-algebra are considered, and the congruence kernel of transitive GE-algebra is characterized. The concept of belligerent GE-filter in GE-algebra is introduced by Bandaru [5]. Relationships between a belligerent GE-filter and a GE-filter are established, and the necessary and sufficient conditions on which a GE-filter to be a belligerent GE-filter are presented. Further, different properties of the union and the product of GE-algebras are investigated.Zadeh, in 1965, studied the notion of fuzzy sets [6]. Following this, Rosenfeld in 1971, used this theory and has formulated the concept of a fuzzy subgroup of a group [7]. Since then, many researchers have been studying several algebraic structures including fuzzy subalgebras, fuzzy subgroups, fuzzy ideals, and fuzzy filters. Jun and Hong in [8] studied the concept of a fuzzy deductive system in Hilbert algebra. Akram and Zham [9] studied sensible fuzzy ideals of BCK-algebras with respect to a t-conorm. Akram and Dar [10] investigated T-fuzzy ideals in BCI-algebras. Additionally, Jun, Akram, and Pasha [11] presented intuitionistic fuzzy quasi-associative ideals in BCI-algebras. Recently, Alemayehu and EShetie [12] studied Fuzzy ideals and fuzzy filters of GE-algebras.Motivated by the above results, in this research, the concept of belligerent fuzzy GE-filter of GE-algebra is introduced. The relationship between a fuzzy GE-filter of GE-algebra and a belligerent fuzzy GE-filter of GE-algebra is given. Further, it is shown that a finite product (union) of belligerent fuzzy GE-filters of GE-algebras is a belligerent fuzzy GE-filter of a finite product (union) of GE-algebras.


## 2. Preliminaries

This section deals on definitions and basic results that we use in the sequel such as the concepts on GE-algebras, GE-filters of GE-algebras, belligerent GE-filters, and fuzzy filters on a given set.Definition 1 (see [13]).
Hilbert algebra is an algebraG,∗,1 of type 2,0 such that the following axioms hold, for all a,b,c∈G.(1)
a∗b∗a=1,(2)
a∗b∗c∗a∗b∗a∗c=1,(3)
ifa∗b=b∗a=1, then a=b.Definition 2 (see [4]).
GE-algebra is a non-empty setG with a constant 1 and a binary operation ∗ satisfying axioms:(1)
a∗a=1,(2)
1∗a=a ,(3)
a∗b∗c=a∗b∗a∗c, for all a,b,c∈G.Definition 3 (see [4]).
GE-algebraG,∗,1 is said to be transitive if it satisfies(1)a∗b≤c∗a∗c∗b,for all a,b,c∈G.Theorem 1 (see [4]).
In a transitive GE-algebraG,∗,1, for all a,b,c∈G, the following conditions hold(1)
a≤bimpliesc∗a≤c∗b(2)
a∗b≤b∗c∗a∗c(3)
a∗b∗b∗c≤a∗c(4)
a≤bandb≤cimpliesa≤c.Definition 4 (see [4]).
GE-algebraG,∗,1 is said to be commutative if it satisfies(2)a∗b∗b=b∗a∗a,for all a,b∈G.Theorem 2 (see [4]).
Every commutative GE-algebra is a generalized Hilbert algebra.Definition 5 (see [4]).
A subsetS of GE-algebra G is called a GE-filter of G if it satisfies the following:(1)
1∈S(2)
ifa∗b∈S and a∈S, then b∈S.Theorem 3 (see [4]).
LetS be a filter of G. If a≤b and a∈S, then b∈S.Theorem 4 (see [4]).
A non-empty subsetS of GE-algebra G is a filter of G if and only if it satisfies 1∈S, a∗b∗c∈S and b∈S implies that a∗c∈S for all a,b,c∈G.Definition 6 (see [5]).
A subsetS of GE-algebra G is called a belligerent GE-filter of G if it satisfies 1∈S and(3)a∗b∗c∈Sanda∗b∈S⇒a∗c∈S,for all a,b,c∈G.Definition 7 (see [5]).
GE-algebraG is said to be left exchangeable if a∗b∗c=b∗a∗c for all a,b,c∈G.Theorem 5 (see [5]).
LetS be a GE-filter of a transitive and left exchangeable GE-algebra G. If S satisfies the condition, a∗a∗b∈S implies that a∗b∈S for all a,b∈G, then S is a belligerent GE-filter of G.Recall that, for any setG a function, ζ:G⟶0,1,∧,∨ is called a fuzzy subset of G [6], where 0,1 is a unit interval, δ∧ν=minδ,ν and δ∨ν=maxδ,ν for all δ,ν∈0,1. A fuzzy subset ν of G is proper if it is a nonconstant function. A fuzzy subset ν such that νa=0 for all a∈G is an improper fuzzy subset. Let ζ:G⟶0,1. For every δ∈0,1, the level subset ζ of G is ζδ=a∈Gδ≤ζa.Definition 8 (see [14, 15]).
Lety∈G,0〈α≤1. A fuzzy point yα of G is a fuzzy subset that is defined as(4)yαb=α,ifb=y,0,otherwise.Definition 9 (see [16]).
A fuzzy subsetζ of a bounded lattice G is said to be a fuzzy ideal of G, if for all a,b∈G,(1)
ζ0=1(2)
ζa∧b≥ζa∨ζb(3)
ζa∨b≥ζa∧ζb.
In [16], Swamy and Raju discussed that, a fuzzy subset ζ of a bounded lattice G is a fuzzy ideal if and only if ζ0=1 and ζa∨b=ζa∧ζb, for all a,b∈G.Theorem 6 (see [16]).
Letζ be a fuzzy subset of G, then ζ is a fuzzy ideal of G if and only if, for any α∈0,1, ζα is an ideal of G.Definition 10 (see [17]).
A fuzzy subsetζ of a bounded lattice G is said to be a fuzzy filter of G, if for all a,b∈G,(1)
ζ1=1(2)
ζa∧b≥ζa∨ζb(3)
ζa∨b≥ζa∧ζb.


## 3. Belligerent Fuzzy GE-Filters

In this topic, the concept of belligerent fuzzy GE-filter of GE-algebra is introduced. The relationship between a fuzzy GE-filter of GE-algebra and a belligerent fuzzy GE-filter of GE-algebra is given.Some concepts of about fuzzy GE-filters of GE-algebras are recalled.Definition 11 (see [12]).
Letζ be a fuzzy subset of a GE-algebra G, then ζ is called a fuzzy GE-filter on G if the following conditions hold for any x,y∈G:(1)
ζ1=1,(2)
ζy≥ζx∗y∧ζx.Lemma 1 (see [12]).
A fuzzy GE-filterζ on a GE-algebra G satisfies the following if x≤y, then ζx≤ζy for all x,y∈G.Theorem 7 (see [12]).
Letζ be a fuzzy subset of G such that ζ1=1 and ζb∗c≥ζa∧ζa∗b∗c. Then, ζ is a fuzzy GE-filter.Definition 12.
LetG be GE-algebra and μ be a fuzzy subset on G, then ζ is said to be a belligerent fuzzy GE-filter on G if for any a,b∈G.(1)
ζ1=1,(2)
ζa∗c≥ζa∗b∗c∧ζa∗b.Example 1.
Consider a setG=a,b,c,d,e,f,1 and define a binary operation ∗ on G as in Table 1.
It is easy to check thatG is GE-algebra. Now, define a fuzzy subset ζ on G by ζ1=1,ζa=ζb=ζf=0.9, and ζc=ζd=ζe=0.8. It is easy to verify that ζ is a belligerent fuzzy GE-filter on G.
The following results show that the relation between belligerent fuzzy GE-filter and fuzzy GE-filter.Table 1
A binary operation∗ defined on a set G.
∗abcdef1a11cee11ba1dddf1c1b11111da1111z1eab11111fabede111abcdef1Theorem 8.
ζ is a belligerent fuzzy GE-filter of GE-algebra G if and only if, ∀α∈0,1, ζα is a belligerent GE-filter.Proof.
Supposeζ be a belligerent fuzzy GE-filter of GE-algebra G, then ζ1=1. This implies 1∈ζ1⊆ζα,whereα∈0,1. Let a,b,c∈G such that a∗b∗c∈ζα and a∗b∈ζα, then ζa∗b∗c≥α and ζa∗b≥α, so that ζa∗c≥ζa∗b∗c∧ζa∗b≥α; thus, a∗c∈ζα. Hence ζα is belligerent GE-filter.
Conversely, suppose thatζα be a belligerent GE-filter of GE-algebra G. Since 1∈ζα for all α∈0,1, 1∈ζ1. Hence, ζ1=1. Let a,b,c∈G such that ζa∗b∗c∧ζa∗b=α for all α∈0,1. Clearly, ζa∗b∗c≥α and ζa∗b≥α. So, a∗b∗c∈ζα and a∗b∈ζα which shows that a∗b∈ζα. Consequently, ζa∗b≥α=ζa∗b∗c∧ζa∗b. Hence ζ is a belligerent fuzzy GE-filter.Corollary 1.
K is a belligerent GE-filter of GE-algebra G if and only if χK is a belligerent fuzzy GE-filter.Theorem 9.
Letζ be a belligerent fuzzy GE-filter of GE-algebra G, then ζ is a fuzzy GE-filter.Proof.
Sinceζ1=1 and ζ is a belligerent fuzzy GE-filter of G, we have ζ1∗b≥ζ1∗a∗b∧ζ1∗a which means ζb≥ζa∗b∧ζa. So, ζ is a fuzzy GE-filter. But the converse of this theorem may not be true.Example 2.
Consider setG and the binary operation ∗ on G given in the above examples (3) and (5). Define a fuzzy GE-filter ζ on G by ζ1=1,ζb=0.8, ζa=ζc=ζd=ζe=ζf=0.6. Since ζd∗f=ζf=0.6≤ζd∗c∗f∧ζd∗c=ζd∗1∧ζ1=ζ1∧ζ1=1∧1=1. This implies μ is not a belligerent fuzzy GE-filter on G.
LetG be GE-algebra, and ζ be any fuzzy subset of G. The following equation is considered as(5)ζa∗b∗a∗c≥ζa∗b∗cforanya,b,c∈G.
Any fuzzy GE-filter cannot satisfy equation (5), so we look at the following examples:Example 3.
Consider the GE-algebraG in Example 1 and fuzzy GE-filters ζ on G as ζ1=1=ζb, ζa=ζc=ζd=ζe=ζf=0.5 does not satisfies equation (1), since ζd∗c∗f=ζd∗1=ζ1=1 and ζd∗c∗d∗f=ζ1∗f=ζf=0.5. Thus ζd∗c∗f≥ζd∗c∗d∗f.
The next result shows that the condition on which a fuzzy GE-filter to be a belligerent fuzzy GE-filter.Theorem 10.
Letζ be a fuzzy GE-filter of GE-algebra G such that ζa∗b∗a∗c≥ζa∗b∗c, then ζ is a belligerent fuzzy GE-filter of G.Proof.
Sinceζ is a fuzzy GE-filter of G, ζ1=1 and ζa∗c≥ζa∗b∗a∗c∧ζa∗b. Also, from the assumption we have ζa∗b∗a∗c∧ζa∗b≥ζa∗b∗c∧ζa∗b so that ζ is a belligerent fuzzy GE-filter of G.Corollary 2.
In belligerent GE-algebra, the trivial fuzzy GE-filter is a belligerent fuzzy GE-filter.GE-algebraG is said to be left exchangeable and transitive if a∗b∗c=b∗a∗c and a∗b≤c∗a∗c∗b for all a,b,c∈G, respectively [4, 5].Theorem 11.
The trivial fuzzy GE-filter of transitive GE-algebraG is a belligerent fuzzy GE-filter.Proof.
Letζ be the trivial fuzzy GE-filter on transitive GE-algebra G. Let x,y,z∈G, then ζx∗z=1=1∧1=ζx∗y∗z∧ζx∗y. Hence, ζ is a belligerent fuzzy GE-filter.
The fuzzy subset given in Theorem7 is not a belligerent fuzzy GE-filter. For if we consider set G and the binary operation ∗ on G given in the above example and define a fuzzy subset ζ on G by ζ1=ζf=1, ζa=ζb=ζc=ζd=ζf=0.7. Since ζc∗b=ζb=0.7≤1=1∧1=ζ1∧ζc∗1=ζc∗d∧ζc∗d∗b. Hence, ζ is not belligerent fuzzy GE-filter on G.Theorem 12.
Letζ be a fuzzy GE-filter on G such that ζa∗b≥ζa∗a∗b. If G is transitive and left exchangeable, then ζa∗b∗a∗c≥ζa∗b∗c.Proof.
Assume thatG is transitive and left exchangeable and ζ be a fuzzy GE-filter on G such that ζa∗b≥ζa∗a∗b. From the transitivity property of G, we get ζb∗c≤ζa∗b∗a∗c and ζa∗b∗c≤ζa∗a∗b∗a∗c. From the left exchangeable property of G, we get ζa∗a∗b∗a∗c=ζa∗a∗a∗b∗c. Now, applying the given condition in the theorem and the left exchangeable property of G yields ζa∗a∗a∗b∗c≤ζa∗a∗b∗c=ζa∗b∗a∗c.Theorem 13.
Letζ be a fuzzy GE-filter of a transitive and left exchangeable GE-algebra G such that ζa∗b≥ζa∗a∗b, then ζ is a belligerent fuzzy GE-filter of G.Proof.
Suppose the conditions in the theorem hold, then, by Theorem12 , we get ζa∗b∗a∗c≥ζa∗b∗c and, hence, from Theorem 10, we conclude that ζ is a belligerent fuzzy GE-filter of G.
For a pointu and a non-empty fuzzy subset ζ of GE-algebra G, define a fuzzy subset ζu on G as(6)ζua=ζu∗afor alla∈G.Lemma 2.
Ifζ is a fuzzy GE-filter of GE-algebra, then ζu1=ζuu=1.Ifζ is a fuzzy GE-filter of G, then fuzzy subset ζu may not be a fuzzy GE-filter.Example 4.
LetG=a,b,c,d,e,f,1 be a GE-algebra of Example 1. A fuzzy subset ζ of G, such as ζ1=ζb=1 and ζa=ζc=ζd=ζe=ζf=0.4, is a fuzzy GE-filter of G but ζd is not a fuzzy GE-filter. Since ζdc∗a=ζd∗c∗a=ζd∗1=ζ1=1, ζdc=ζd∗c=ζ1=1, and ζda=ζd∗a=ζa=0.4. This implies ζdc∗a∧ζdc〉ζda. Thus, ζd is not a fuzzy GE-filter of G.
However, we give the conditions thatζd is a fuzzy GE-filter of G in the following result.Theorem 14.
Letζ be a belligerent fuzzy GE-filter on G, then ζu is a fuzzy GE-filter on G.Proof.
Letu,a,b∈G. Since ζ is a belligerent fuzzy GE-filter on G, we obtain that ζu∗b≥ζu∗a∗b∧ζu∗a, i.e., ζub≥ζua∗b∧ζua. So, ζu is a fuzzy GE-filter on G.
We suggest the conditions under which a fuzzy GE-filter can be a belligerent fuzzy GE-filter.Theorem 15.
Letζ be a fuzzy subset on G, ζ1=1 and for any u∈G, ζu be a fuzzy GE-filter on G. Then, ζ is a belligerent fuzzy GE-filter on G.Proof.
Sinceζu is a fuzzy GE-filter on G, for all a∈G, we get(7)ζac≥ζab∗c∧ζab.So that ζa∗c≥ζa∗b∗c∧ζa∗b. Thus, ζ is a belligerent fuzzy GE-filter on G.Corollary 2.
Letζ be a fuzzy GE-filter and ζu be a fuzzy GE-filter of G for every u∈G, then ζ is a belligerent fuzzy GE-filter on G.Theorem 16.
Letζ be a belligerent fuzzy GE-filter on G, then fuzzy subset ζu, where u∈G is the smallest fuzzy GE-filter on G such that ζ⊆ζu.Proof.
From Theorem14, ζu is a fuzzy GE-filter on G. For any u,a∈G, we obtain that a≤u∗a and hence, ζa≤ζu∗a=ζua. Hence ζ⊆ζu. Let η be a fuzzy GE-filter on G such that ζ⊆η and ηu=1. Hence ζ⊆ζu. Now, for any a∈G, ζua=ζu∗a⊆ηu∗a=ηu∗a∧ηu≤ηa. This implies ζu is the least fuzzy GE-filter of G.Theorem 17.
Leth:G1⟶G2 be a GE-morphism from GE-filters G1 to G2 and ζ be a belligerent fuzzy GE-filter of G2, then h−1ζ is a belligerent fuzzy GE-filter of G1.Proof.
Suppose the conditions hold. Sinceh−1ζa=ζha for all a∈G1 and h is a morphism, h−1ζ1=ζh1=ζ1 and(8)h−1ζa∗b∧h−1μa=ζha∗b∧ζha=ζha∗hb∧ζha≤ζhb=h−1ζb.
Hence,h−1ζ is a fuzzy GE-filter of G1. Also,(9)h−1ζa∗c=ζha∗c=ζhx∗hz≥ζhx∗hy∗hz∧ζhx∗hy=ζha∗hb∗c∧ζha∗b=ζha∗hb∗c∧ζhx∗y=ζhx∗y∗z∧ζhx∗y=h−1ζa∗b∗c∧h−1ζa∗b,so that h−1ζ is a belligerent fuzzy GE-filter of G1.


## 4. Direct Product of Finite Belligerent Fuzzy GE-Filters of GE-Algebras

In this section, it is discussed that the finite product (union) of belligerent fuzzy GE-filters of GE-algebras becomes a belligerent fuzzy GE-filter of the finite product (union) of GE-algebras. Further the sufficient and necessary conditions on which the finite product of a belligerent GE-filter of a finite product of GE-algebras becomes a belligerent fuzzy GE-filter is given.Definition 13.
Letζ1:G1⟶0,1 and ζ2:G2⟶0,1 be fuzzy subsets of GE-algebras G1 and G2, respectively. Then the direct product of fuzzy subsets ζ1 and ζ2 is denoted by ζ1×ζ2:G1×G2⟶0,1 and is defined as(10)ζ1×ζ2a1,a2=ζ1a1∧ζ2a2,for any a1∈G1 and a2∈G2.Definition 14.
Letζ1,ζ2,ζ3,…,ζn be n fuzzy subsets of GE-algebras G1,G2,G3,…,Gn, respectively. Then the direct product of finite fuzzy subsets of GE-algebras is denoted by ζ1×ζ2×ζ3×⋯×ζn and is defined as(11)ζ1×ζ2×ζ3×⋯×ζn:G1×G2×G3×⋯×Gn⟶0,1,ζ1×ζ2×⋯×ζna1,a2,a3,…,an=ζ1a1∧ζ2a2∧…∧ζnan for any a1∈G1,a2∈G2,…,an∈Gn.Theorem 18.
Letζ1,ζ2,ζ3,…,ζn be n (belligerent) fuzzy GE-filters of GE-algebras G1,G2,G3,…,Gn, respectively. Then ζ1×ζ2×ζ3×⋯×ζn is (belligerent) fuzzy GE-filter of G1×G2×G3×⋯×Gn.Proof.
Suppose that the assumptions hold and leta1,b1,c1∈G1;a2,b2,c2∈G2,…,an,bn,cn∈Gn. Define a binary operation ⊛ on G1×G2×⋯×Gn by(12)a1,a2,,…,an⊛b1,b2,…,bn=a1∗1b1,a2∗2b2,…,an∗nbn,where ∗1,∗2,…,∗n are the binary operations on G1,G2,…,Gn, respectively. Since ζi is fuzzy GE-filter for each i,1≤i≤n, it is trivial to show that ζ1×ζ2×⋯×ζna1,a2,…,an≥ζ1×ζ2×⋯×ζnb1,b2,…,bn⊛a1,a2,…,an∧ζ1×ζ2×⋯×ζnb1,b2,…,bn. From the given hypothesis ζi is a belligerent fuzzy GE-filter for each i,1≤i≤n. So,(13)ζ1a1∗1b1≥ζ1a1∗1c1∗1b1∧ζ1a1∗1c1,ζ2a2∗2b2≥ζ2a2∗2c2∗2b2∧ζ2a2∗2c2,⋅⋅⋅ζnan∗nbn≥ζnan∗ncn∗nbn∧ζnan∗ncn.
After some steps it follows that(14)ζ1×ζ2×⋯×ζna1,a2,…,an⊛b1,b2,…,bn≥ζ1×ζ2×⋯×ζna1,a2,…,an⊛c1,c2,…,cn⊛b1,b2,…,bn∧ζ1×ζ2×⋯×ζna1,a2,…,an⊛c1,c2,…,cn.
Therefore,ζ1×ζ2×⋯×ζn is a belligerent fuzzy GE-filter of G1×G2×⋯×Gn.Example 5.
LetG1=1,a,b,c,d and G2=1,a,b,c,d,e be GE-algebras where the binary operations ∗1 and ∗2 are defined by Tables 2 and 3.
Then,G1×G2=1,1,1,a,1,b,1,c,1,d,1,e,a,1,a,e,a,a,a,b,a,c,a,d,b,1,b,e,b,a,b,b,b,c,b,d,c,1,c,e,c,a,c,b,c,c,c,d,d,1,d,e,d,a,d,b,d,c,d,d is GE-algebra with point wise operation.
Now, defineζ1:G1⟶0,1 by ζ11=ζ1b=1,ζ1a=ζ1c=ζ1d=0.3 and ζ2:G2⟶0,1 by ζ21=ζ2e=ζ2a=1,ζ2b=ζ2c=ζ2d=0.4. Clearly, ζ1 and ζ2 are fuzzy GE-filters of G1 and G2, respectively. Now, we define ζ1×ζ2:G1×G2⟶0,1 by ζ1×ζ21,1=ζ1×μ21,e=ζ1×ζ21,a=ζ1×ζ2b,1=ζ1×ζ2b,e=ζ1×ζ2b,a=1,(15)ζ1×ζ21,b=ζ1×ζ21,c=ζ1×ζ21,d=ζ1×ζ2b,b=ζ1×ζ2b,c=ζ1×ζ2b,d=0.4.
ζ1×ζ2a,1=ζ1×ζ2a,e = ζ1×ζ2a,b=ζ1×ζ2a,c=ζ1×ζ2a,d = ζ1×ζ2c,1=ζ1×ζ2c,e=ζ1×ζ2c,b = ζ1×ζ2c,c=ζ1×ζ2c,d=ζ1×ζ2d,1=ζ1×ζ2d,e = ζ1×ζ2d,b=ζ1×ζ2d,c=ζ1×ζ2d,d=0.3. Then, clearly, ζ1×ζ2 is a fuzzy GE-filter of G1×G2.Table 2∗11wxyz11wxyzw11x11x1z1yzy1wx1wz11x11Table 3
A binary operation∗ 2 defined on a set G2.
∗21rwxyz11rwxyzr111xyzw1r1yyyx11w111y1r1111z1r1111Theorem 19.
Ifζ1 and ζ2 are (belligerent) fuzzy GE-filters of G1 and G2, respectively, then the union ζ1∪ζ2 is a (belligerent) fuzzy GE-filter of G1∪G2.Proof.
Clearly,ζ1∪ζ21=ζ11∨ζ21=1∨1=1. Recall that for any a,b∈G1∪G2, define a binary operation ∗ on G1∪G2 as a∗b=a∗1b,ifa,b∈G1,a∗2b,ifa,b∈G2,bifaandbare not belong to the same GE−algebra. Now,(16)ζ1∪ζ2b=ζ1b∨ζ2b≥ζ1a∗b∧ζ1a∨ζ2a∗b∧ζ2a=ζ1∨ζ2a∗b∧ζ1∨ζ2a=ζ1∪ζ2a∗b∧ζ1∪ζ2a.
Hence,ζ1∪ζ2 is a fuzzy GE-filter of G1∪G2. Next, assume that ζ1 and ζ2 be belligerent fuzzy GE-filters of G1 and G2, respectively. Then(17)ζ1∪ζ2a∗c=ζ1a∗1c∨ζ2a∗2c≥ζ1a∗1b∗1c∧ζ1a∗1b∨ζ2a∗2b∗2c∧ζ2a∗2b=ζ1∨ζ2a∗b∗c∧ζ1∨ζ2a∗b=ζ1∪ζ2a∗b∗c∧ζ1∪ζ2a∗b,so that ζ1∪ζ2 is a belligerent fuzzy GE-filter of G1∪G2.Theorem 20.
Letζ1,ζ2,ζ2,…,ζnbenfuzzy subsets of GE-algebrasG1,G2,G3,…,Gn, respectively, and lett∈0,1, thenζ1×ζ2×ζ2×⋯×ζnt=ζ1t×ζ2t×ζ2t×⋯×ζnt.Proof.
Lett∈0,1 and for any element(18)a1,a2,a3,…,an∈ζ1×ζ2×ζ3×⋯×ζnt⇔ζ1×ζ2×ζ3×⋯×ζna1,a2,a3,…,an≥t⇔ζ1a1∧ζ2a2∧ζ3a3∧…∧ζnan≥t⇔ζ1a1≥t,ζ2a2≥t,ζ3a3≥t,…,ζnan≥t⇔a1∈ζ1t,a2∈ζ2t,a3∈ζ3t,…,an∈ζnt⇔a1,a2,a3,…,an∈ζ1t×ζ2t×ζ3t×⋯×ζnt.
Henceζ1×ζ2×ζ2×⋯×ζnt=ζ1t×ζ2t×ζ2t×⋯×ζnt.Theorem 21.
Letζ1,ζ2,ζ2,…,ζn be n fuzzy subsets of GE-algebra G1,G2,G3,…,Gn, respectively. Then ζ1×ζ2×ζ2×⋯×ζn is a fuzzy GE-filter of a GE-algebra G1×G2×G3×⋯×Gn if and only if for any t∈0,1, ζ1×ζ2×ζ2×⋯×ζnt is a GE-filter of a GE-algebra G1×G2×G3×⋯×Gn.Proof.
Suppose thatζ1×ζ2×ζ2×⋯×ζn is a fuzzy GE-filter of GE-algebra G1×G2,G3×⋯×Gn. Since ζ1×ζ2×ζ2×⋯×ζn1=ζ11∧ζ21∧ζ21∧…∧ζn1=1. This implies 1∈ζ1×ζ2×ζ2×⋯×ζnt.
Ifa∗b∈ζ1×ζ2×ζ2×⋯×ζnt and a∈ζ1×ζ2×ζ2×⋯×ζnt, then ζ1×ζ2×ζ2×⋯×ζna∗b≥t, ζ1×ζ2×ζ2×⋯×ζna≥t. This implies ζ1×ζ2×ζ2×⋯×ζna∗b∧ζ1×ζ2×ζ2×⋯×ζna≥t. This implies ζ1×ζ2×ζ2×⋯×ζnb≥t. Thus, y∈μ1×μ2×ζ3×⋯×ζnt. Thus, ζ1×ζ2×ζ3×⋯×ζn is a GE-filter of G1×G2×G3×⋯×Gn.
Conversely, suppose thatζ1×ζ2×ζ3×⋯×ζnt is a GE-filter. Clearly ζ1×ζ2×ζ3×⋯×ζn1=1. Let ζ1×ζ2×ζ3×⋯×ζna∗b∧ζ1×ζ2×ζ3×⋯×ζna=t. This implies ζ1×ζ2×ζ3×⋯×ζna∗b≥t,ζ1×ζ2×ζ3×⋯×ζna≥t. This implies a∗b∈ζ1×ζ2×ζ2×⋯×ζnt and a∈ζ1×ζ2×ζ2×⋯×ζnt. Thus, b∈ζ1×ζ2×ζ2×⋯×ζnt. This implies ζ1×ζ2×ζ3×⋯×ζnb≥t=ζ1×ζ2×ζ3×⋯×ζna∗b∧ζ1×ζ2×ζ3×⋯×ζna.
Thus,ζ1×ζ2×ζ3×⋯×ζna∗b∧ζ1×ζ2×ζ3×⋯×ζna is a fuzzy GE-filter.Theorem 22.
Letζ1,ζ2,ζ2,…,ζn be n fuzzy subsets of GE-algebra G1,G2,G3,…,Gn, respectively. Then, ζ1×ζ2×ζ3×⋯×ζn is a belligerent fuzzy GE-filter of GE-algebra G1×G2×G3×⋯×Gn if and only if for any t∈0,1, ζ1×ζ2×ζ2×⋯×ζnt is belligerent GE-filter of GE-algebra G1×G2×G3×⋯×Gn.Proof.
The proof of it is straight forward by Theorems8 and 18.


## 5. Conclusion and Future Work

In this paper, the concept of belligerent fuzzy GE-filters is introduced. We investigate the relationships between fuzzy GE-filter and a belligerent fuzzy GE-filter. Additionally prove that that the finite product (union) of belligerent fuzzy GE-filters of GE-algebras becomes a belligerent fuzzy GE-filter of the finite product (union) of GE-algebras. We hope in the future, we study prominent fuzzy GE-filters, imploring fuzzy GE-filters, soft GE-filter, and Soft GE-ideals of GE-algebras.



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