Strong fuzzy GE-filters and fuzzy GE-ideals of bordered GE-algebras

Teferi Getachew Alemayehu; Workaferahu Solomon Eshetie; Ravikumar Bandaru; Young Bae Jun

doi:10.1515/taa-2023-0104

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Strong fuzzy GE-filters and fuzzy GE-ideals of bordered GE-algebras

Teferi Getachew Alemayehu , Workaferahu Solomon Eshetie , Ravikumar Bandaru and Young Bae Jun

Published/Copyright: December 1, 2023

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From the journal Topological Algebra and its Applications Volume 11 Issue 1

Abstract

The notion of strong fuzzy generalized exchange (GE)-filters and fuzzy GE-ideals of bordered GE-algebras is studied, and their related properties are investigated. We proved that the intersection of strong fuzzy GE-filters (fuzzy GE-ideals) is a strong fuzzy GE-filter (fuzzy GE-ideal). We studied the conditions for a fuzzy subset of a bordered GE-algebra to be a strong fuzzy GE-filter and the characterization of a strong fuzzy GE-filter. We studied that under some conditions, any fuzzy subset of a bordered GE-algebra can be a fuzzy GE-ideal. A given fuzzy GE-ideal and an element in a bordered fuzzy GE-algebra is established.

Keywords: GE-algebras; bordered GE-algebras; fuzzy GE-filters; fuzzy GE-ideals

MSC 2010: 06F35; 03G25; 08A30

1 Introduction

Imai and Iseki [13] studied BCK-algebras as the algebraic semantics for a nonclassical logic possessing only implication. Next, many scholars studied the generalized concepts of BCK-algebras. Kim and Kim [15] presented the concepts of a BE-algebra as a generalization of a dual BCK-algebra. Hilbert algebras were presented by Henkin and Skolem in the fifties for studies in intuitionistic and other nonclassical logics. Diego [12] showed that Hilbert algebras form a variety that is locally finite. Rezaei et al. [18] discussed the relationships between Hilbert algebras and BE-algebras, i.e., generalization process in the study of algebraic structures is also an important area of study. As a generalization of Hilbert algebras, Bandaru et al. [10,11,16] studied the concept of GE-algebras and studied several properties such as properties of strong GE-filters and GE-ideals in bordered GE-algebras.

Zadeh [24] developed the idea of fuzzy sets to address vagueness. Since then, studies on the idea of fuzzy sets have picked up steam in a variety of fields, including engineering, artificial intelligence, medicine, signal processing, and expert systems. To segment medical images, Kaur and Chaira [14] investigated a unique fuzzy technique. A novel measure of divergence is proposed by Verma and Maheshwari [21] and applied to multi-criteria decision-making in a fuzzy environment. Fuzzy generalized prioritized weighted average operator and its use in multiple attribute decision-making by Verma and Sharma [22].

Additionally, abstract algebra successfully and conveniently integrates fuzzy set theory. A fuzzy subgroup of a group is described by Rosenfeld [19]. Later, a lot of algebraists became interested in introducing fuzzy theory in various algebraic structures by fuzzifying the formal theory: in fuzzy ideals and fuzzy filters of MS-algebras [4–6,8], in fuzzy ideals and fuzzy filters of BL-algebras [15,20,23], in fuzzy ideals of universal algebras [1–3], in fuzzy ideals of Ockham algebras [7], etc. Moreover, recently, Bandaru et al. [11] studied fuzzy GE-filters of GE-algebras.

Initiated by all the above results, in this article, the concept strong fuzzy GE-filters and fuzzy GE-ideals of a bordered GE-algebras are studied, and their related properties are investigated. We prove that the intersection of strong fuzzy GE-filters (resp., fuzzy GE-ideals) is a strong fuzzy GE-filter (resp., fuzzy GE-ideal). However, we have used examples to show that the union of strong fuzzy GE-filters (resp., fuzzy GE-ideals) is generally not a strong fuzzy GE-filters (resp., fuzzy GE-ideal). We studied the conditions for a fuzzy subset of a bordered GE-algebra to be a strong fuzzy GE-filter and the characterization of a strong fuzzy GE-filter.

Finally, we studied under which conditions any fuzzy subset of a bordered GE-algebra can be a fuzzy GE-ideal.

2 Preliminaries

In this topic, we recall the basic results used to the main result.

Definition 2.1

[10] An algebra ( X , ∗ , 1 ) of type ( 2 , 0 ) is said to be a GE-algebra if it satisfies the following axioms:

s ∗ s = 1 ,
1 ∗ s = s ,
s ∗ ( t ∗ u ) = s ∗ ( t ∗ ( s ∗ u ) ) , for all s , t , u ∈ X .

In a GE-algebra ( X , ∗ , 1 ) , construct a binary relation “ ≤ ” on X by s ≤ t iff s ∗ t = 1 .

Theorem 2.2

[10] Let X be a GE-algebra. Then, the following conditions hold for all s , t , u ∈ X :

s ∗ 1 = 1 ,
s ∗ ( s ∗ t ) = s ∗ t ,
1 ≤ s ⇒ s = 1 ,
s ≤ t ∗ s ,
s ≤ ( s ∗ t ) ∗ t ,
s ≤ ( t ∗ s ) ∗ t ,
s ≤ ( s ∗ t ) ∗ s ,
s ≤ t ∗ ( t ∗ s ) ,
s ∗ ( t ∗ u ) ≤ t ∗ ( s ∗ u ) ,
s ≤ t ∗ u ⇔ t ≤ s ∗ u .

Theorem 2.3

[10] Let ( X , ∗ , 1 ) be a GE-algebra. Then, for any s , t , u ∈ X , the following are equivalent.

s ∗ t ≤ ( u ∗ s ) ∗ ( u ∗ t ) ,
s ∗ t ≤ ( t ∗ u ) ∗ ( s ∗ u ) .

Definition 2.4

[10] Let a ( X , ∗ , 1 ) be GE-algebra. Then, X is said to be transitive if it satisfies s ∗ t ≤ ( u ∗ s ) ∗ ( u ∗ t ) for any s , t , u ∈ X .

Theorem 2.5

[10] Let ( X , ∗ , 1 ) be a transitive GE-algebra. Then, for any s , t , u ∈ X , the following properties are true:

s ≤ t ⇒ u ∗ s ≤ u ∗ t ,
s ∗ t ≤ ( t ∗ u ) ∗ ( s ∗ u ) ,
s ≤ t ⇒ t ∗ u ≤ s ∗ u ,
( ( s ∗ t ) ∗ t ) ∗ u ≤ s ∗ u ,
s ≤ t and t ≤ u ⇒ s ≤ u .

Definition 2.6

[10] An algebra ( X , ∗ , 1 ) is said to be commutative GE-algebra if ( s ∗ t ) ∗ t = ( t ∗ s ) ∗ s , for any s , t ∈ X .

Definition 2.7

[10] A nonempty subset F of GE-algebra X is said to be a filter if:

1 ∈ F ,
s ∗ t ∈ F and s ∈ F , then t ∈ F .

Definition 2.8

[9] A GE-algebra X is called bordered GE-algebra if it has a special element 0, which satisfies 0 ≤ s for all s ∈ X .

For every element x of a bordered GE-algebra X , we denote x * 0 by x 0 , and ( x 0 ) 0 is denoted by x 00 .

Theorem 2.9

[9] Let X be a bordered GE-algebra. Then, for any s , t ∈ X , the following properties are true:

1 0 = 0 , 0 0 = 1 ,
s ≤ s 00 , 0 ≤ s 00 ,
0 ≤ s ∗ s 0 = s 0 ,
s ∗ t 0 ≤ t ∗ s 0 ,
s ≤ t 0 ⇔ t ≤ s 0 ,
s ∗ t 0 = x ∗ ( t ∗ s 0 ) .
Let X be a transitive bordered GE-algebra. Then,
s ≤ t ⇔ t 0 ≤ s 0 ,
s ∗ t ≤ t 0 ∗ s 0 .
Let X be an antisymmetric bordered GE-algebra. Then,
s ∗ t 0 = t ∗ s 0 .
Let X be a transitive and antisymmetric bordered GE-algebra. Then,
s 000 = s 0 .

Definition 2.10

[9] Let X be a bordered GE-algebra. s ∈ X is called a duplex bordered element if s 00 = s .

0 2 ( X ) represents the set of all duplex bordered elements of a bordered GE-algebra X . It is said to be the duplex bordered set of X .

Definition 2.11

[9] An algebra X is said to be a duplex bordered GE-algebra if it is a bordered GE-algebra and every element is duplex, i.e, X = 0 2 ( X ) .

Let X be a bordered GE-algebra and F ⊆ X . We define the set 0 2 ( X , F ) = { s ∈ X : s 00 ∈ F } .

Definition 2.12

[16] A GE-filter F of a bordered GE-algebra X is said to be strong if it satisfies 0 2 ( X , F ) ⊆ F , i.e., s ∈ F , whenever s 00 ∈ F .

Definition 2.13

[16] A subset A of a bordered GE-algebra X is said to be a GE-ideal of X if:

0 ∈ A ,
( ( t 0 ∗ s 0 ) 0 ) ∈ A , t ∈ A ⇒ s ∈ A for any s , t ∈ X .

Recall that, for each set A , a function μ : A → [ 0 , 1 ] is called a fuzzy subset of A . For each α ∈ [ 0 , 1 ] , the set

μ α = { s ∈ A : μ ( x ) ≥ α }

is called the level subset of μ at α [24].

A fuzzy set μ in a set X of the form

μ ( r ) = α ∈ ( 0 , 1 ] , if s = r , 0 , s ≠ r

is said to be a fuzzy point with support s and value α and is denoted by s α .

For a fuzzy set μ in a set X and α ∈ ( 0 , 1 ] , we say that a fuzzy point s α is

Contained in μ , denoted by s α ∈ μ , [17] if μ ( s ) ≥ α .
Quasi-coincident with μ , denoted by s α q μ , [17] if μ ( a ) + α ≥ 1 .

We remember, in [11], given α ∈ ( 0 , ] and a fuzzy set μ in Ge-algebra X , we consider the following sets:

( μ , α ) ∈ = s ∈ X : x y ∈ μ and

( μ , α ) q = s ∈ X : x y q μ ,

which are called an ∈ α - s e t and q α - s e t of μ , respectively.

Definition 2.14

[11] A fuzzy set μ in GE-algebra X is called a fuzzy GE-filter of X if it satisfies:

(2.1) ( ∀ α ∈ ( 0 , 1 ] ) ( ( μ , α ) ∈ ≠ ∅ ⇒ 1 ∈ ( μ , α ) ∈ ) ,

(2.2) x ∗ y ∈ ( μ , α a ) ∈ , x ∈ ( μ , α b ) ∈ ⇒ y ∈ ( μ , min { α a , α b } ) ∈ ,

for all x , y ∈ X and α a , α b ∈ ( 0 , 1 ] .

Theorem 2.15

[11] A fuzzy subset λ of GE-algebra X is called a fuzzy GE-filter of X if and only if:

(2.3) μ ( 1 ) ≥ μ ( s ) , ∀ s ∈ X ,

(2.4) μ ( t ) ≥ μ ( s ∗ t ) ∧ μ ( s ) , ∀ s , t ∈ X .

Theorem 2.16

[11] For any fuzzy GE-filter μ of GE-algebra X , μ ( s ) ≤ μ 0 ( t ) , whenever s ≤ t .

3 Strong fuzzy GE-filters of bordered GE-algebras

Let μ be a fuzzy subset of a bordered GE-algebra X . We define the fuzzy subset 0 2 ( μ ) by 0 2 ( μ ) ( u ) = μ ( u 00 ) for all u ∈ X .

Definition 3.1

Let X be a bordered GE-algebra. A fuzzy GE-filter μ of X is said to be a strong fuzzy GE-filter if it satisfies

(3.1) u ∈ ( 0 2 ( μ ) , α ) ∈ ⇒ u ∈ ( μ , α ) ∈

for all u ∈ X and α ∈ ( 0 , 1 ] .

Example 3.2

Let X = { 0 , 1 , s , t , u } be a set with a binary operation ∗ given in the following table:

Then, X is a bordered GE-algebra [16]. Define μ : X → [ 0 , 1 ] as μ ( 1 ) = μ ( s ) = μ ( t ) = 1 and μ ( u ) = μ ( 0 ) = 0.5 . It is routine to verify that μ is a strong fuzzy GE-filter of X .

∗	0	1	s	t	u
0	1	1	1	1	1
1	0	1	s	t	u
s	0	1	1	1	0
t	u	1	1	1	u
u	1	1	1	t	1

Theorem 3.3

Every fuzzy GE-filter in a duplex bordered GE-algebra is a strong fuzzy GE-filter.

Proof

Let X be a duplex bordered GE-algebra and μ be fuzzy GE-filter of X . Let u ∈ ( 0 2 ( μ ) , t ) ∈ . Then, t ≤ 0 2 ( μ ) ( u ) = μ ( u 00 ) = μ ( u ) . This implies that u ∈ ( μ , α ) ∈ . Hence, μ is a strong fuzzy GE-filter.□

Theorem 3.4

A fuzzy GE-filter μ in a bordered GE-algebra is a strong fuzzy GE-filter if and only if it satisfies

(3.2) o 2 ( μ ) ⊆ μ .

Proof

Suppose that a fuzzy GE-filter μ of a bordered GE-algebra is a strong fuzzy GE-filter. Let α = 0 2 ( μ ) ( u ) for α ∈ ( 0 , 1 ] and u ∈ X . Then, u ∈ ( 0 2 ( μ ) , α ) ∈ and so u ∈ ( μ , α ) ∈ . This implies that 0 2 ( μ ) ( u ) = α ≤ μ ( u ) . Hence, o 2 ( μ ) ⊆ μ .

Conversely, suppose that a fuzzy GE-filter μ satisfies o 2 ( μ ) ⊆ μ .

Let u ∈ ( o 2 ( μ ) , α ) ∈ for any α ∈ ( 0 , 1 ] and u ∈ X . Then, α ≤ ( o 2 ( μ ) ) ( u ) ≤ μ ( u ) . Thus, u ∈ ( μ , α ) ∈ . Hence, μ is a strong fuzzy GE-filter.□

Theorem 3.5

The intersection of two strong fuzzy GE-filters of a bordered GE-algebra X is strong fuzzy GE-filter.

Proof

Let μ and λ be strong fuzzy GE-filters of a bordered GE-algebra X . By [11], μ ∩ λ is a fuzzy GE-filter of X . Let t ∈ ( 0 , 1 ] and u ∈ ( μ ∩ λ ) ∈ . Then, 0 2 ( μ ∩ λ ) ( u ) = ( μ ∩ λ ) ( u 00 ) = μ ( u 00 ) ∧ λ ( u 00 ) ≤ ( μ ( u ) ∧ λ ) ( u ) = ( μ ∩ λ ) ( u ) . Therefore, u ∈ ( μ ∩ λ , t ) ∈ Hence, μ ∩ λ is a strong fuzzy GE-filter of X .□

However, the union of strong fuzzy GE-filters may not be a strong fuzzy GE-filter.

Example 3.6

In the following table, define a binary operation ∗ in X = { 0 , 1 , s , t , u , v , } as follows, which is a bordered GE-algebra

∗	0	1	s	t	u	v	w
0	1	1	1	1	1	1	1
1	0	1	s	t	u	v	w
s	t	1	1	t	t	1	1
t	s	1	t	1	v	v	w
u	s	1	s	1	1	1	w
v	0	1	s	t	t	1	w
w	0	1	s	t	u	v	1

Define a function ξ , η on X as η ( 1 ) = η ( w ) = η ( v ) = η ( s ) = 1 and η ( t ) = η ( u ) = η ( 0 ) = 0.5 and ξ ( 1 ) = ξ ( t ) = ξ ( u ) = ξ ( v ) = ξ ( w ) = 1 and ξ ( s ) = ξ ( 0 ) = 0.6 . It is routine to verify that ξ and η are strong fuzzy GE-filters of X . But ( η ∪ ξ ) ( s ∗ 0 ) ∧ ( η ∪ ξ ) ( s ) = 1 and ( η ∪ ξ ) ( 0 ) = 0.6 . This implies that ( η ∪ ξ ) ( s ∗ 0 ) ∧ ( η ∪ ξ ) ( s ) > ( η ∪ ξ ) ( 0 ) . Hence, η ∪ ξ is not a GE-filter of X . Therefore, η ∪ ξ is not a strong fuzzy GE-filter.

Theorem 3.7

A fuzzy GE-filter λ in X is a strong fuzzy GE-filter of X if and only if the nonempty ∈ α -set ( μ , α ) ∈ of μ in X is a strong GE-filter of X for all α ∈ ( 0 , 1 ] .

Proof

Suppose that λ in X is a strong fuzzy GE-filter of X . By [11], ∈ α -set ( μ , α ) ∈ of μ in X is a GE-filter of X . Let u ∈ ( 0 2 ( λ ) , α ) ∈ . Then, α ≤ 0 2 ( λ ) ( u ) ≤ λ ( u ) . Thus, u ∈ ( μ , α ) ∈ . Hence, ∈ α -set ( μ , α ) ∈ of μ in X is a strong GE-filter of X for all α ∈ ( 0 , 1 ] .

Conversely, suppose that nonempty ∈ α -set ( μ , α ) ∈ of μ in X is a strong GE-filter of X for all α ∈ ( 0 , 1 ] . Then ( μ , α ) ∈ is a fuzzy GE-filter of X by Theorem 4.13 of [11]. Let α ∈ ( 0 , 1 ] be such that α = 0 2 ( λ ) ( u ) . Then, u ∈ ( 0 2 ( λ ) , α ) ∈ . Since ( λ , α ) ∈ is a strong GE-filters of X , u ∈ ( λ , α ) ∈ . Thus, λ ( u ) ≥ α = 0 2 ( λ ) ( u ) . Thus, λ is a strong fuzzy GE-filter of X .□

Lemma 3.8

[16] Let X be a transitive and antisymmetric bordered GE-algebra. Then, for any u , v , w ∈ X

( v ∗ ( u ∗ w ) = u ∗ ( v ∗ w ) = ( u ∗ v ) ∗ ( u ∗ w ) ) ,
u 00 = u 0 ∗ u ,
u ≤ u 0 ∗ v , v ≤ u 0 ∗ v ,
u + v = u 00 + v 00 ,
u ∗ ( u ∗ v ) 00 = ( u ∗ v ) 00 ,
( u ∗ v ) 00 = u 00 ∗ v 00 ,

where u + v = ( u ∗ v 0 ) 0

Theorem 3.9

Let X be a transitive and antisymmetric bordered GE-algebra. A fuzzy subset μ of X satisfies

For any u , v ∈ X such that u ≤ v and μ ( u ) ≤ μ ( v ) ,
0 2 ( μ ) ⊆ μ ,
μ ( u + v ) ≥ min { μ ( u ) , μ ( v ) } for any u , v ∈ X ,

then μ is a strong fuzzy GE-filter of X .

Proof

Suppose that conditions ( 1 ) , ( 2 ) , and ( 3 ) are satisfied. We prove that μ is a strong fuzzy GE-filter of X . To prove it, enough to show, λ is a fuzzy GE-filter of X .

For any u , u ≤ 1 , and μ ( u ) ≤ μ ( 1 ) by (1).

In [16] u ≤ u 00 and u ∗ v ≤ ( u ∗ v ) 00 , for any u , v ∈ X .

min { μ ( u ) , μ ( u ∗ v ) } ≤ min { μ ( u 00 ) , μ ( ( u ∗ v ) 00 ) } by ( 1 ) ≤ { μ ( u 00 + ( u ∗ v ) 00 ) } by ( 3 ) = { μ ( u 00 ∗ ( u ∗ v ) 0 ) 0 } ≤ { μ ( v 00 ) } after routine work ≤ { μ ( v ) } by ( 2 ) .

Thus, μ is a fuzzy GE-filter of X by Theorem 2.15. Therefore, μ is a strong fuzzy GE-filter of X .□

Theorem 3.10

If μ is a strong fuzzy GE-filter of X, then the nonempty Q α -set ( μ , α ) q of μ is a strong GE-filter of X for all α ∈ ( 0 , 1 ] .

Proof

By [11] Q t -set ( μ , α ) q of μ is a GE-filter of X for all α ∈ ( 0 , 1 ] .

Let u ∈ ( 0 2 ( μ ) , α ) ∈ for all α ∈ ( 0 , 1 ] , for all u ∈ X . Then, α ≤ ( 0 2 ( μ ) ( u ) ) = μ ( u ) . Thus, u ∈ ( μ ) , α ) ∈ .

Hence, ( μ , α ) q of μ is a strong GE-filter of X .□

4 Fuzzy GE-ideals of a bordered GE-algebra

Definition 4.1

A fuzzy set λ in X is called a fuzzy GE-ideal of X if it satisfies:

(4.1) ( ∀ t ∈ ( 0 , 1 ] ) ( λ , α ) ∈ ≠ ∅ ⇒ 0 ∈ ( λ , α ) ∈

(4.2) ( v 0 ∗ u 0 ) 0 ∈ ( λ , α a ) ∈ , v ∈ ( λ , α b ) ∈ , ⇒ u ∈ ( λ , min { α a , α b } ) ,

for all u , v ∈ X and α a , α b ∈ ( 0 , 1 ] .

Example 4.2

Let X = { 0 , 1 , a , b , c , d } be a set with a binary operation ∗ , given as follows:

∗	0	1	a	b	c	d
0	1	1	1	1	1	1
1	0	1	a	b	c	d
a	d	1	1	1	d	d
b	c	1	a	1	c	c
c	b	1	a	b	1	1
d	b	1	a	b	1	1

Then, X = { 0 , 1 , a , b , c , d } is a bordered GE-algebra [16]. Define μ : X → [ 0 , 1 ] as μ ( 0 ) = μ ( a ) = μ ( b ) = 0.9 and μ ( 1 ) = μ ( c ) = μ ( d ) = 0.6 and λ : X → [ 0 , 1 ] as λ ( 0 ) = 0.8 , λ ( c ) = λ ( d ) = 0.7 , and λ ( a ) = λ ( b ) = λ ( 1 ) = 0.3 . It is routine to verify that μ and λ are the fuzzy GE-ideals of X .

Theorem 4.3

A fuzzy subset λ of a bordered GE-algebra X is a fuzzy GE-ideal of X if and only if it satisfies

(4.3) λ ( 0 ) ≥ λ ( u ) , for all u ∈ X ,

(4.4) λ ( u ) ≥ min { λ ( v ) , λ ( ( v 0 ∗ u 0 ) 0 ) } , for all u , v ∈ X .

Proof

Suppose that λ is a fuzzy GE-filter of X . Assume that there exists a ∈ X such that λ ( 0 ) ≤ λ ( a ) . Let α 0 = λ ( a ) . Then, a ∈ ( λ , α 0 ) ∈ . This implies ( λ , α 0 ) ∈ ≠ ∅ . Thus, 0 ∈ ( λ , α 0 ) ∈ , i.e., λ ( 0 ) ≥ α 0 , which is a contradiction. Hence, λ ( 0 ) ≥ λ ( u ) for all u ∈ X .

Let u , v ∈ X be such that α a = λ ( v ) and α b = λ ( ( v 0 ∗ u 0 ) 0 ) . Then, v ∈ ( λ , α a ) ∈ and ( v 0 ∗ u 0 ) 0 ∈ ( λ , α b ) ∈ . Since λ is a fuzzy GE-ideal of X , we have u ∈ ( λ , min { α a , α b } ) ∈ . Hence, λ ( u ) ≥ min { α a , α b } = min { λ ( v ) , λ ( ( v 0 ∗ u 0 ) 0 ) } .

Thus, a fuzzy subset λ of a bordered GE-algebra X is a fuzzy GE-ideal of X , which satisfies 4.3 and 4.4.

Converse, assume that μ satisfies 4.3 and 4.4. Let α ∈ ( 0 , 1 ] and u ∈ ( λ , α ) ∈ . Then, λ ( u ) ≥ α , and hence, λ ( 0 ) ≥ λ ( u ) ≥ α . Thus, 0 ∈ ( λ , α ) ∈ . Let u , v ∈ X be such that ( v 0 ∗ u 0 ) 0 ∈ ( λ , α a ) ∈ , v ∈ ( λ , α b ) ∈ . Then, λ ( ( v 0 ∗ u 0 ) 0 ) ≥ α a and λ ( v ) ≥ α b . Therefore, λ ( u ) ≥ min { λ ( ( v 0 ∗ u 0 ) 0 ) , λ ( v ) ) ≥ min { α a , α b } by 4.4. Hence, u ∈ ( λ , min { α a , α b } ) ∈ . Thus, λ is a fuzzy GE-ideal of X .□

Theorem 4.4

If μ and λ are fuzzy GE-ideals of a bordered GE-algebra X, then, so is μ ∩ λ .

Proof

Suppose that μ and λ are fuzzy GE-ideals of a bordered GE-algebra X . Then, ( μ ∩ λ ) ( 0 ) = min { μ ( 0 ) , λ ( 0 ) } ≥ min { μ ( 0 ) , λ ( 0 ) } ≥ min { μ ( x ) , λ ( x ) } .

Next,

( μ ∩ λ ) ( u ) = min { μ ( u ) , λ ( u ) } ≥ min { min { μ ( ( v 0 ∗ u 0 ) 0 ) , μ ( v ) } , min { λ ( ( v 0 ∗ u 0 ) 0 ) , λ ( v ) } } ≥ min { min { μ ( ( v 0 ∗ u 0 ) 0 ) , λ ( ( v 0 ∗ u 0 ) 0 ) } , min { λ ( v ) , λ ( v ) } } = min { ( μ ∩ λ ) ( ( v 0 ∗ u 0 ) 0 ) , ( μ ∩ λ ) ( v ) } .

Hence, μ ∩ λ is fuzzy GE-ideals of a bordered GE-algebra X .□

But the union of fuzzy GE-ideals may not be a fuzzy GE-ideal of bordered GE-algebra.

Example 4.5

In Example 4.2, the union λ ∪ μ of μ and λ is given as follows: ( μ ∪ λ ) ( 0 ) = ( μ ∪ λ ) ( a ) = ( μ ∪ λ ) ( b ) = 0.9 , ( μ ∪ λ ) ( c ) = ( μ ∪ λ ) ( d ) = 0.7 , and ( μ ∪ λ ) ( 1 ) = 0.6 . But λ ∪ μ is not a fuzzy GE-ideal of X , since ( b 0 ∗ 1 0 ) 0 ∈ ( λ ∪ μ , 0.7 ) ∈ and b ∈ ( λ ∪ μ , 0.7 ) ∈ , but 1 ∉ ( λ ∪ μ , 0.7 ) ∈ .

Theorem 4.6

Let X be a bordered GE-algebra. Every fuzzy GE-ideal λ of X satisfies

(4.5) ( ∀ u , v ∈ X and ∀ α ∈ ( 0 , 1 ] ) ( v ∈ ( λ , α ) ∈ , v 0 ≤ u 0 ⇒ u ∈ ( λ , α ) ∈ ) .

Proof

Suppose that v 0 ≤ u 0 , then v 0 ∗ u 0 = 1 . This implies that ( v 0 ∗ u 0 ) 0 = 1 0 = 0 . Since λ is fuzzy GE-ideal of X , ( u 0 ∗ v 0 ) 0 ∈ ( λ , α ) ∈ , and v ∈ ( λ , α ) ∈ . Hence, u ∈ ( λ , α ) ∈ .□

Corollary 4.7

Every fuzzy GE-ideal λ of a transitive bordered GE-algebra X satisfies

(4.6) ( ∀ u , v ∈ X and ∀ α ∈ ( 0 , 1 ] ) ( v ∈ ( λ , α ) ∈ , v 0 ≤ u 0 ⇒ u ∈ ( λ , α ) ∈ ) .

Proof

Since X is a transitive bordered GE-algebra, for all u , v ∈ X , u ≤ v implies v 0 ≤ u 0 . By Theorem 4.6, the result holds.□

Theorem 4.8

A fuzzy GE-ideal λ of a transitive bordered GE-algebra X satisfies

(4.7) ( u ∗ v ) 0 ∈ ( λ , α a ) ∈ , v ∈ ( λ , α b ) ∈ ⇒ u ∈ ( λ , min { α a , α b } ) ∈ ,

for all u , v ∈ X , for all α a , α b ∈ ( 0 , 1 ] .

Proof

Since X is a transitive bordered GE-algebra, ( u 0 ∗ y 0 ) 0 ≤ ( u ∗ v ) 0 . Also, λ is a fuzzy GE-ideals of X , and by Corollary 4.7, we have ( u 0 ∗ v 0 ) 0 ∈ ( λ , α a ) ∈ . Hence, u ∈ ( λ , min { α a , α b } ) ∈ .□

Theorem 4.9

If a bordered GE-algebra X is an antisymmetric and duplex, then every fuzzy subset λ of X satisfying λ ( 0 ) ≥ λ ( u ) and λ ( u ) ≥ λ ( v ) ∧ λ ( ( u ∗ v ) 0 ) for all u , v ∈ X is fuzzy GE-ideal of X.

Proof

We prove that λ ( u ) ≥ λ ( v ) ∧ λ ( ( v 0 ∗ u 0 ) 0 ) for all u , v ∈ X .

Since X is a bordered and duplex GE-algebra, u ∗ v = v 0 ∗ u 0 . λ ( v ) ∧ λ ( ( v 0 ∗ u 0 ) 0 ) = λ ( v ) ∧ λ ( ( u ∗ v ) 0 ) ≤ λ ( u ) for all u , v ∈ X . Hence, λ is a fuzzy GE-ideals of X .□

We explore the conditions under which the ∈ α -set and Q α -set can be GE-filters.

Theorem 4.10

Given a fuzzy set λ in X, its ∈ t -set ( λ , α ) ∈ is a GE-ideal of X for all α ∈ ( 0.5 , 1 ] if and only if λ satisfies:

(4.8) ( ∀ u ∈ X ) ( λ ( u ) ≤ max { λ ( 0 ) , 0.5 } ) ,

(4.9) ( ∀ u , v ∈ X ) ( min { λ ( v ) , λ ( ( v 0 ∗ v 0 ) 0 } ) ≤ max { λ ( u ) , 0.5 } ) .

Proof

Assume that the ∈ α -set ( λ , α ) ∈ is a GE-ideal of X for all t ∈ ( 0.5 , 1 ] . If there exists a ∈ X such that α = λ ( a ) ∈ ( 0.5 , 1 ] and λ ( a ) ≰ max { λ ( 0 ) , 0.5 } , then a α ∈ λ and 0 α ∉ λ , i.e., a ∈ ( λ , α ) ∈ and 0 ∈ ( λ , α ) ∈ . This is a contradiction, and thus, λ ( u ) ≤ max { λ ( 0 ) , 0.5 } for all u ∈ X .

If 4.9 is not valid, then min { λ ( b ) , λ ( ( b 0 ∗ a 0 ) 0 ) } > max { λ ( a ) , 0.5 } for some a , b ∈ X . If we take α = min { λ ( b ) , λ ( ( b 0 ∗ a 0 ) 0 ) } , then α ∈ ( 0.5 , 1 ] , b α , ( b 0 ∗ a 0 ) 0 α ∈ λ . Hence, b ∈ ( λ , α ) ∈ , and b ∈ ( λ , α ) ∈ , ( b 0 ∗ a 0 ) 0 ∈ ( λ , α ) ∈ , which imply that a ∈ ( λ , α ) ∈ . Thus, λ ( a ) ≥ α > 0.5 , which is a contradiction. Therefore, min { λ ( v ) , λ ( ( v 0 ∗ v 0 ) 0 } ) ≤ max { λ ( u ) , 0.5 } for all u , v ∈ X .

Conversely, suppose that λ satisfies 4.8 and 4.9. Let ( λ , α ) ∈ ≠ ∅ for all α ∈ ( 0.5 , 1 ] . Then, there exists a ∈ ( λ , α ) ∈ , and thus, a α ∈ λ . It follows from 4.8 that 0.5 ≤ α ≤ λ ( a ) ≤ max { λ ( 0 ) , 0.5 } . Thus, 0 ∈ ( λ , α ) ∈ . Let α ∈ ( 0.5 , 1 ] and u , v ∈ X be such that v ∈ ( λ , α ) ∈ and ( v 0 ∗ u 0 ) 0 ∈ ( λ , α ) ∈ . Then, u α ∈ λ and ( v 0 ∗ u 0 ) 0 α ∈ λ , i.e., λ ( v ) ≥ α and λ ( ( v 0 ∗ v 0 ) 0 ) ≥ α . Using 4.9, we obtain

max { λ ( u ) , 0.5 } ≥ min { λ ( v ) , λ ( ( v 0 ∗ u 0 ) 0 ) } ≥ α ≥ 0.5 and so u α ∈ λ . i.e., u ∈ ( λ , α ) ∈ . Therefore, ( λ , α ) ∈ is a GE-ideal of X for all α ∈ ( 0 , 1 ] .□

Theorem 4.11

A fuzzy set λ in X is a fuzzy GE-ideal of X if and only if the nonempty ∈ α -set ( λ , α ) ∈ of λ in X is a GE-ideal of X for all α ∈ ( 0 , 1 ] .

Proof

Suppose that a fuzzy set λ in X is a fuzzy GE-ideal of X . Let α ∈ ( 0 , 1 ] be such that ( λ , α ) ∈ ≠ ∅ . Then, there exists a ∈ ( λ , α ) ∈ . It follows that α ≤ λ ( a ) ≤ λ ( 0 ) . Hence, 0 ∈ ( λ , α ) ∈ . Let u , v ∈ X be such that v ∈ ( λ , α ) ∈ and ( v 0 ∗ u 0 ) 0 ∈ ( λ , α ) ∈ . Then, α ≤ λ ( v ) and α ≤ λ ( ( v 0 ∗ u 0 ) 0 ) , which implies that α ≤ min { λ ( v ) , λ ( ( v 0 ∗ u 0 ) 0 ) } ≤ λ ( u ) . Hence, u ∈ ( λ , α ) ∈ . This implies that ( λ , α ) ∈ is a GE-ideal of X .

Conversely, assume that the nonempty ∈ α -set ( λ , α ) ∈ of λ in X is a GE-ideal of X for all α ∈ ( 0 , 1 ] . If λ ( 0 ) ≤ λ ( a ) for some a ∈ X , then a ∈ ( λ , λ ( a ) ) ∈ and 0 ∉ ( λ , λ ( a ) ) ∈ . This is contradiction, and thus λ ( 0 ) ≥ λ ( u ) for all u ∈ X . If there exists a , b ∈ X such that λ ( a ) < min { λ ( b ) , λ ( ( b 0 ∗ a 0 ) 0 ) } , then b ∈ ( λ , α ) ∈ and ( b 0 ∗ a 0 ) 0 ∈ ( λ , α ) ∈ , but a ∈ ( λ , α ) ∈ for α = min { λ ( b ) , λ ( ( b 0 ∗ a 0 ) 0 ) } . This is a contradiction, hence λ ( u ) ≥ min { λ ( v ) , λ ( ( v 0 ∗ u 0 ) 0 ) } for all u , v ∈ X . Therefore, λ is a fuzzy GE-filter of X by Theorem 4.3.□

Theorem 4.12

If λ is a fuzzy GE-ideal of X, then the nonempty Q α -set ( λ , α ) q of λ is a GE-ideal of X for all α ∈ ( 0 , 1 ] .

Proof

Let λ be a fuzzy GE-ideal of X , and assume that ( λ , α ) q ≠ ∅ for all ( α , 1 ] . Then, there exists a ∈ ( λ , α ) q and so a α q λ , i.e., λ ( a ) + α ≥ 1 . Hence, λ ( 0 ) + α ≥ λ ( a ) + α ≥ 1 . Thus, 0 ∈ ( λ , α ) q . Let u , v ∈ X be such that v ∈ ( λ , α ) q and ( v 0 ∗ u 0 ) 0 ∈ ( λ , α ) q . Then, v α q λ and ( v 0 ∗ u 0 ) 0 α q λ , i.e., λ ( v ) + α ≥ 1 and λ ( ( v 0 ∗ u 0 ) 0 ) + α ≥ 1 . It follows that

λ ( u ) + α ≥ min { λ ( v ) , λ ( ( v 0 ∗ u 0 ) 0 ) } + 1 ≥ min { λ ( v ) , λ ( ( v 0 ∗ u 0 ) 0 ) + 1 } ≥ 1 .

Hence, u α q λ , and therefore, u ∈ ( λ , α ) q . Consequently, ( λ , α ) q is a GE-ideal of X for all α ∈ ( 0 , 1 ] .□

5 Conclusion and future work

In this article, we discussed the ideas of strong fuzzy GE-filters and fuzzy GE-ideals, as well as their associated features. We show that the intersection of strong fuzzy GE-filters (fuzzy GE-ideals) is a strong fuzzy GE-filter (fuzzy GE-ideal). We a gave criteria under which a fuzzy subset of a bordered GE-algebra to be a strong fuzzy GE-filter. Also, we suggested what conditions under which any fuzzy subset of a bordered GE-algebra can be a fuzzy GE-ideal. In the future, we will study promoting fuzzy GE-filters and imploring fuzzy GE-filters of GE-algebras.

Also, we will study neutrosophic GE-algebera, neutrosophic GE-filter, and their properties.

Acknowledgments

Authors would like to thank the referee for his valuable comments and suggestions to improve this presentation.

Funding information: The author states no funding involved.
Conflict of interest: The authors declare that there are no conflicts of interest regarding the publication of this article.
Data availability statement: This study is not supported by any data.

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Received: 2022-12-31

Accepted: 2023-11-07

Published Online: 2023-12-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/taa-2023-0104

Keywords for this article

GE-algebras; bordered GE-algebras; fuzzy GE-filters; fuzzy GE-ideals

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