F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.180502.1 Research Article Articles Fuzzy Ideals and Fuzzy Filters on Implication Algebras

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 Derso Derebew Nigussie Conceptualization Software Validation Writing – Review & Editing https://orcid.org/0000-0003-2431-2802 a 1 Dejen Gerima Tefera Conceptualization Data Curation Software Supervision Validation Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0000-0002-9033-560X 2 Mathematics, Woldia University, Woldia, Amhara, 400, Ethiopia Mathematics, Wollo University, Dessie, Amhara, Ethiopia nderebew@gmail.com

No competing interests were disclosed.

 12 5 2026 2026 15 723 25 4 2026 Copyright: © 2026 Derso DN and Dejen GT 2026 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Background

In this paper, we present the intersection of a family of fuzzy ideals and fuzzy filters as fuzzy ideals and fuzzy filters, respectively. We introduced the concept of fuzzy implication algebras as a fuzzification of implication algebras.

 Method

Within this framework, we further define and study fuzzy implication ideals, fuzzy implication filters, and fuzzy normal subalgebras, each equipped with membership functions that satisfy the compatibility conditions reflecting the underlying implication operation. These fuzzy notions are presented as appropriate generalizations of classical concepts of ideals, subalgebras, and filters.

 Result

We establish several foundational results for these constructions and prove different characterization theorems. The characterization results provide several equivalent descriptions, such as those expressed through internal algebraic conditions, order-theoretic constraints, and implication-based inequalities, thereby clarifying when a given fuzzy subset qualifies as fuzzy implication ideal, filter, or normal subalgebra.

 Conclusion

Consequently, the theory yields a unified and systematic method for verifying and constructing fuzzy-algebraic structures.

 Implication algebra B-Algebra BCK- Algebra fuzzy ideal fuzzy filters and fuzzy sets The author(s) declared that no grants were involved in supporting this work. 1. Introduction

Ever since LA.Zadeh introduced the concept of a fuzzy set, the philosophy behind this idea has pre- meated various disciplines of human knowlwdge including those of logic and reasoning which is the foundation stone of all Mathematical Sciences in. 14

Xu et al. 11 proposed the concept of lattice implication algebra and discussed some of its properties in. 9 Xu and Qin in 10 introduced the idea of a filter and an implicative filter in a lattice implica- tion algebra and investigated their properties, and Kim,C.B. and Kim,H.S. in 5 initaited related ideas on BM algebras.

Xu et al. 11 provided some equivalent conditions for a filter to be an implicative filter in a lattice implication algebra. Yong Bae Jun 12 fuzzified the concept of positive implicative filters and alternative filters in lattice implication algebras. 13

Abbott 1 introduced basic ideas on orthoimplication algebras, and Gerima 4 initiated the concepts of ideals and filters on implication algebras. Roh and et al in 6 discussed on some important prop- erties on lattice implication algebras, in addition Neggers and et al in 7 discussed on basic ideals of B-algebra, and more important properties of ideals in an implication algebras has been investigated in, 9 and Ravi Kumar Bandaru, and et al, introduced the notion of falling fuzzy implicative filter of a BE −algebra, relations between fuzzy implicative filters, and falling fuzzy implicative filters in, 8 and Berhanu, and et al in 2 initiated the idea of Almost Distributive Fuzzy Lattice based on principal ideal fuzzy lattice.The Concepts of Hilbert Implication algebra and generalized Hilbert Implicationalgebr was introduced in. 3

Finally, in this paper, the fuzzification of implication algebras, implication ideals, and implication filters are extended to fuzzy ideals in implication algebra,and fuzzy implication filters with different additional properties.

 2. Methods and materials

Every fuzzy normal subset μ in A is a fuzzy implication algebra.

The converse of this theorem doesnot hold as illustrated by the following example.

Fuzzy implication algerbra.
⇒ 1 a b c
1 1 a b C
a 1 1 1 1
b 1 a 1 1
c 1 a b 1

Let A = {1, a, b, c, d }, defined by the table 2 below:

( A, ⇒, 1) is an implication algebra. So that we have μ ( b ) = μ ( 1 ⇒ b ) = μ ( ( a ⇒ a ) ⇒ ( c ⇒ b ) ) ≥ μ ( a ⇒ c ) ∧ μ ( a ⇒ b ) = μ ( 1 ) ∧ μ ( 1 ) = μ ( 1 ) .

Hence μ( b) ≥ μ(1) is not true.

Fuzzy normal subset.
⇒ 1 a b C d
1 1 a b C d
a 1 1 1 1 1
b 1 a 1 1 1
c 1 a b 1 1
d 1 a b C 1

Let B be a non empty set . Then a fuzzy subset μ in B is a function μ : B → [0, 1] .

4 If ( B, ⇒, 1) an implication algebra, then For any a, b ∈ B, a ⇒ b = { 1 , if a ≤ b b , if a > b

4 Let ( B, ⇒, 1) be an implication algebra. Then a non-empty subset S of an implication algebra B is called a sub algebra of B if a, b ∈ S, then a ⇒ b ∈ S.

4 A nonempty subset I of an implication algebra B is called an implication ideal of B if the following condition holds:

0 ∈ I

a ⇒ b ∈ I and b ∈ I , imply a ∈ I ,a, b ∈ A.

4 A nonempty subset F of an implication algebra B is called an implication filter if the following condition holds:

1 ∈ F

a ⇒ b ∈ F and a ∈ F , imply b ∈ F .

3. Main Results 3.1 Fuzzy implication algebra

A fuzzy subset μ in A is called a fuzzy implication algebra if it satisfies the inequality μ ( a ⇒ b ) ≥ μ ( a ⇒ b ) ∧ μ ( b ) , a , b ∈ A .

Let A = {1, a, b, c} be a set defined by the table 1 below and a < b < c < 1 .

Define a fuzzy subset μ : A → [0, 1] by μ(1) = μ( b) = 0.8 and μ( x) = 0.1 , ∀ x ∈ A|{1, b} . Then μ( a ⇒ b) ≥ µ ( a ⇒ b ) ∧ µ ( b ) = µ ( 1 ) ∧ µ ( b ) = 0.8 ∧ 0.8 = 0.8 .

Hence μ( a ⇒ b) ≥ 0.8 .

Therefore μ is a fuzzy implication algebra.

Every fuzzy implication algebra μ satisfies the inequality μ(1) ≥ μ( a) , ∀ a ∈ A.

Let ( A, ⇒, 1) be an implication algebra and let a ∈ A be any element in A. µ ( 1 ) = µ ( a ⇒ a ) ≥ µ ( a ⇒ a ) ∧ µ ( a ) ≥ µ ( 1 ) ∧ µ ( a ) = µ ( a ) .

Hence μ(1) ≥ μ( a) , ∀ a ∈ A.□

If a fuzzy subset μ in A is a fuzzy implication algebra, then the following holds: F A 1 : µ ( a ⇒ 1 ) ≥ µ ( a ) . F A 2 : µ ( a ⇒ ( b ⇒ 1 ) ) = µ ( 1 ) , ∀ a , b ∈ A .

Let ( A, ⇒, 1) be implication algebra and let a,b be any element in A.

Since a ⇒ 1 = 1 , we have μ( a ⇒ 1) = μ(1) ≥ μ( a) by theorem 3.2. Hence μ( a ⇒ 1) ≥ μ( a) , ∀ a ∈ A.

Therefore F A1 holds.

Let a, b ∈ A. Then μ( a ⇒ ( b ⇒ 1)) ≥ μ( a ⇒ ( b ⇒ 1) ∧ μ( b ⇒ 1)) .

But a ⇒ ( b ⇒ 1) = a ⇒ 1 = 1 and b ⇒ 1 = 1 . So that we get μ( a ⇒ ( b ⇒ 1)) ∧ μ(1) = μ(1) . Hence μ( a ⇒ ( b ⇒ 1)) ≥ μ(1)…(1) .

But μ(1) ≥ μ( c) ,for c = a ⇒ ( b ⇒ 1) ∈ A. We get μ(1) ≥ μ( a ⇒ ( b ⇒ 1))…(2) . Thus μ( a ⇒ ( b ⇒ 1) = μ(1) , ∀ a, b, 1 ∈ A.□

Let ( A, ⇒, 1) be an implication algebra and let μ be a fuzzy subset of A. Then the α− level cut of μ is μ α = { a ∈ A| μ( a) ≥ α} ,α ∈ [0, 1] .

A fuzzy subset μ of an implication algebra A is Fuzzy implication algebra if and only if μ α, α ∈ [0, 1] is an implication sub algebra.

Suppose μ is a fuzzy subset of A and μ(1) ≥ α,α ∈ [0, 1] . Imply that 1 ∈ μ α. Hence μ α ≠ ø .

Let a, b ∈ μ α . Then μ( a) ≥ α and μ( b) ≥ α. Since a ≤ a ⇒ b imply that μ( a) ≤ μ( a ⇒ b) , As a result

μ( a ⇒ b) ≥ μ( a) ≥ α. Imply that μ( a ⇒ b) ≥ α. So that we get a ⇒ b ∈ μ α. Hence μ α is an implication sub algebra.

Conversely, suppose μ α is an implication sub algebra. Let a, b ∈ μ α. Then a ⇒ b ∈ μ α, Since μ α is an implication s As a result we get μ( a ⇒ b) ≥ α.

Let μ( a) = α andμ( b) = α . We get α ≤ μ( a ⇒ b) . Put α = μ( a ⇒ b) ∧ μ( b) . So that we get μ( a ⇒ b) ≥ μ( a ⇒ b) ∧ μ( b) , ∀, a, b ∈ A. Hence the result.□

If a fuzzy subset μ in A satisfy F A1 and F A2 , then μ is a fuzzy implication algebra.

Let ( A, ⇒, 1) be an implication algebra. Then a fuzzy subset μ in A is said to be fuzzy normal if it satisfies the inequality μ(( x ⇒ a) ⇒ ( y ⇒ b)) ≥ μ( x ⇒ y ) ∧ μ( a ⇒ b) , ∀ a, b, x, y ∈ A.

Let ( A, ⇒, 1) be implication algebra. Then define a fuzzy subset μ : A → [0, 1] by μ(1) = μ( a) = 0.9 and μ( b) = μ( c) = 0.4 in example 3.1. So μ((1 ⇒ b) ⇒ ( a ⇒ c) = μ( b ⇒ 1) = μ(1) = 0.9 (1) .

μ(1 ⇒ a) ∧ μ( b ⇒ c) = μ( a) ∧ μ(1) = 0.9 ∧ 0.9 = 0.9. (2) .

Hence μ(1 ⇒ b) ⇒ ( a ⇒ c) ≥ μ(1 ⇒ a) ∧ μ( b ⇒ c) by (1) and (2). Therefore μ is a fuzzy normal implication algebra.

ut it holds only for μ( b) = μ(1) , since μ(1) ≥ μ( a) , ∀ a ∈ A. Therefore the converse is not ingeneral true.

If a fuzzy subset μ in A is a fuzzy normal implication algebra, then μ( a ⇒ b) = μ( b ⇒ , ∀ a, b ∈ A.

Let μ be a fuzzy normal subset of A, and let a, b ∈ A. Then, µ ( a ⇒ b ) = µ ( 1 ⇒ ( a ⇒ b ) ) = µ ( ( b ⇒ b ) ⇒ ( a ⇒ b ) ) ≥ µ ( b ⇒ a ) ∧ µ ( b ⇒ b ) = µ ( b ⇒ a ) ∧ µ ( 1 ) = µ ( b ⇒ a )

Hence µ ( a ⇒ b ) ≥ µ ( b ⇒ a ) , ∀ a , b ∈ A … ( 1 )

Again, µ ( b ⇒ a ) = µ ( 1 ⇒ ( b ⇒ a ) ) ≥ µ ( a ⇒ b ) ∧ µ ( a ⇒ a ) = µ ( a ⇒ b ) ∧ µ ( 1 ) = µ ( a ⇒ b ) .

Hence µ ( b ⇒ a ) ≥ µ ( a ⇒ b ) , ∀ a , b ∈ A ( 2 )

Therefore µ ( a ⇒ b ) = µ ( b ⇒ a ) , ∀ a , b ∈ A . □ = µ ( ( a ⇒ a ) ⇒ ( b ⇒ a ) )

Let μ be a fuzzy normal implication algebra. Then the set A μ = { a ∈ A| μ( a) = μ(1)} is a normal implication subalgebra of A.

Let μ be fuzzy normal implication algebra. Thus, it is sufficient to show A μ is normal.

Let a, b, x, y ∈ A such that x ⇒ y ∈ A μ and a ⇒ b ∈ A μ. Then μ( x ⇒ y ) = μ( a ⇒ b) = μ(1). Because μ is fuzzy normal, we have µ ( ( x ⇒ a ) ⇒ ( y ⇒ b ) ) ≥ µ ( x ⇒ y ) ∧ µ ( a ⇒ b ) = µ ( 1 ) ∧ µ ( 1 ) = µ ( 1 )

Hence μ( x ⇒ a) ⇒ ( y ⇒ b) ≥ μ(1)….(1). But μ(1) ≥ μ( z), z = ( x ⇒ a) ⇒ ( y ⇒ b). Imply that μ(1) ≥ μ(( x ⇒ a) ⇒ ( y ⇒ b))…(2).

Hence μ(( x ⇒ a) ⇒ ( y ⇒ b)) = μ(1) by (1) and (2). Therefore ( x ⇒ a) ⇒ ( y ⇒ b) ∈ A μ

Thus A μ is normal.□

The intersection of any set of fuzzy normal implication algebra is also a fuzzy normal implication algebra.

Let { μ α | α ∈ Λ} be a family of fuzzy normal implication algebra, and let a, b, x, y ∈ A. Then ( ∩ α ∈ Λ µ α ) ( ( x ⇒ a ) ⇒ ( y ⇒ b ) ) = Inf α ∈ Λ µ α ( ( x ⇒ a ) ⇒ ( y ⇒ b ) ) ≥ inf α ∈ Λ { min { µ α ( x ⇒ y ) , µ α ( a ⇒ b ) } } = min { inf α ∈ Λ µ α ( x ⇒ y ) , inf α ∈ Λ µ α ( a ⇒ b ) } = min { ( ∩ α ∈ Λ µ α ) ( x ⇒ y ) , ( ∩ α ∈ Λ ) µ α ( a ⇒ b ) }

Hence ∩ α∈Λ μ α is a fuzzy normal set in A. Consequently, ∩ α∈Λ μ α is a fuzzy normal implication algebra.

A fuzzy subset μ in an implication algebra A is called a fuzzy implication ideal of A if it satisfies:

μ(0) ≥ μ( a) , ∀ a ∈ A.

μ( a) ≥ μ( a ⇒ b) ∧ μ( b) , ∀ a, b ∈ A.

Let A = {0, 1, 2, 3} be a set defined by the table 3 below:

So that ( A, ⇒, 3) is an implication algebra.

Define a fuzzy subset μ in A by μ(0) = 0.9 , μ(1) = μ(2) = μ(3) = 0.5 . So that the following holds

μ(0) ≥ μ( a) , ∀ a ∈ A|{0} .

μ( a ⇒ b) ≥ μ( a ⇒ b) ∧ μ( b) , ∀ a, b ∈ A. Hence μ is a fuzzy implication ideal of A.

Fuzzy ideal.
⇒ 0 1 2 3
0 3 1 2 3
1 0 3 2 3
2 0 1 3 3
3 0 1 2 3

A fuzzy subset μ is a fuzzy implication ideal of an implicative algebra A if and only if μ t is an implication ideal of A, t ∈ [0, 1] .

Assume that fuzzy subset μ is a fuzzy implication ideal of A. Since μ(0) ≥ μ( a), ∀ a ∈ A. We have 0 ∈ μ t , t ∈ [0, 1].

Hence μ t ≠ ø.

Let a, b ∈ μ t . Then μ( a) ≥ t and μ( b) ≥ t . Because μ is a fuzzy implication ideal, we have μ( a ⇒ b) ≥. μ( a ⇒ b) ∧ μ( b) ≥ t , which imply that a ⇒ b ∈ μ t . So that a ⇒ b ∈ μ t and b ∈ μ t imply a ∈ μ t .

Hence μ t is an implication ideal of A.

Conversely, suppose μ t is an implication ideal of A. Let a, b ∈ μ t . Then, μ( a) ≥ t and μ( b) ≥ t for t ∈ [0, 1] and a, b ∈ A.

Consequently μ t , implies that μ( a ⇒ b) ≥ t . Put t = μ( a ⇒ b) ∧ μ( b). Hence μ( a ⇒ b) ≥ μ( a ⇒ b) ∧ μ( b), ∀ a, b ∈ A.

Again, for t ∈ [0, 1] and 0 ∈ μ t , μ(0) ≥ t . where t = μ( a) and t = μ( b). Therefore, μ(0) ≥ μ( a) = μ( b). Hence μ(0) ≥ μ( a), ∀ a ∈ A.

Hence μ is a fuzzy implication ideal of A.

A fuzzy subset μ in an implication fuzzy algebra A is a fuzzy implication filter if the following condition holds:

μ(1) ≥ μ( a) , ∀, a ∈ A.

μ( a ⇒ b) ≥ μ( a) ∧ μ( a ⇒ b) , ∀ a, b ∈ A.

Let A = {1, a. b, c} be a set defined by the table 4 below:

Trivially ( A, ⇒, 1) is an implication algebra. Define a fuzzy subset μ of A by μ(1) = 0.8, and μ( a) = μ( b) =

μ( c) = 0.6 . So that we have

μ(1) ≥ μ( a) , ∀, a, b ∈ A|{1} .

μ( a ⇒ b) ≥ μ( a) ∧ μ( a ⇒ b) , ∀ a, b ∈ A. Hence μ is a fuzzy implication filter of A.

A fuzzy subset μ of an implication algebra A is a fuzzy implication filter if and only if

μ t is an implication filter of A.

The intersection of family of Fuzzy Filters is also Fuzzy Filter.

Fuzzy filters.
⇒ 1 a b c
1 1 a b c
a 1 1 b c
b 1 1 1 c
c 1 1 b 1
4. Conclusion

In this paper, the concepts of Ideals and filters in implication algebra are extended to fuzzy ideals and fuzzy filters of implication algebras.

In addition, fuzzy normal ideal in an implication algebra, fuzzy implication sub-algebras, and intersection of fuzzy ideal and fuzzy filters are investigated, and different characterizations are discussed. In future work, the authors will extend this idea to intuitionistic fuzzy implication algebras and other related theories.

 Data availability

No datasets were generated or analyzed during this study. All results are derived analytically, and all supporting information is fully contained within the manuscript.

 Acknowledgments

The authors thank the referees for their comments.

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