Fuzzy Ideals and Fuzzy Filters on Implication Algebras

Derebew Nigussie Derso; Gerima Tefera Dejen

doi:10.12688/f1000research.180502.1

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Research Article

Fuzzy Ideals and Fuzzy Filters on Implication Algebras

[version 1; peer review: awaiting peer review]

Derebew Nigussie Derso ¹, Gerima Tefera Dejen²

PUBLISHED 12 May 2026

Author details Author details

¹ Mathematics, Woldia University, Woldia, Amhara, 400, Ethiopia
² Mathematics, Wollo University, Dessie, Amhara, Ethiopia

Derebew Nigussie Derso
Roles: Conceptualization, Software, Validation, Writing – Review & Editing

Gerima Tefera Dejen
Roles: Conceptualization, Data Curation, Software, Supervision, Validation, Writing – Original Draft Preparation, Writing – Review & Editing

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Abstract

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.

Keywords

Implication algebra,B-Algebra, BCK- Algebra,fuzzy ideal, fuzzy filters,and fuzzy sets

Corresponding author: Derebew Nigussie Derso

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

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

How to cite: Derso DN and Dejen GT. Fuzzy Ideals and Fuzzy Filters on Implication Algebras [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:723 (https://doi.org/10.12688/f1000research.180502.1) First published: 12 May 2026, 15:723 (https://doi.org/10.12688/f1000research.180502.1) Latest published: 12 May 2026, 15:723 (https://doi.org/10.12688/f1000research.180502.1)

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 in¹⁰ 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. in⁵ 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 Jun¹² fuzzified the concept of positive implicative filters and alternative filters in lattice implication algebras.¹³

Abbott¹ introduced basic ideas on orthoimplication algebras, and Gerima⁴ initiated the concepts of ideals and filters on implication algebras. Roh and et al in⁶ discussed on some important prop- erties on lattice implication algebras, in addition Neggers and et al in⁷ 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 in² 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

Theorem 2.1

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.

Table 1. 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

Example 2.1

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

(A, ⇒, 1) is an implication algebra. So that we have

\begin{matrix} μ (b) = μ (1 \Rightarrow b) \\ = μ ((a \Rightarrow a) \Rightarrow (c \Rightarrow b)) \\ \geq μ (a \Rightarrow c) \land μ (a \Rightarrow b) \\ = μ (1) \land μ (1) = μ (1) . \end{matrix}

Hence μ(b) ≥ μ(1) is not true.

Table 2. 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

Definition 2.2

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

Propositions 2.2

⁴If (B, ⇒, 1) an implication algebra, then For any a, b ∈ B, $a \Rightarrow b = {\begin{array}{c} 1, & if a \leq b \\ b, & if a > b \end{array}$

Definition 2.3

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.

Definition 2.4

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

1. 0 ∈ I
2. a ⇒ b ∈ I and b ∈ I , imply a ∈ I ,a, b ∈ A.

Definition 2.5

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

1. 1 ∈ F
2. a ⇒ b ∈ F and a ∈ F , imply b ∈ F .

3. Main Results

3.1 Fuzzy implication algebra

DefiniJion 3.1

A fuzzy subset μ in A is called a fuzzy implication algebra if it satisfies the inequality

μ (a \Rightarrow b) \geq μ (a \Rightarrow b) \land μ (b), a, b \in A .

Example 3.1

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) ≥

\begin{array}{l} µ (a \Rightarrow b) \land µ (b) \\ = µ (1) \land µ (b) = 0.8 \land 0.8 = 0.8 . \end{array}

Hence μ(a ⇒ b) ≥ 0.8.

Therefore μ is a fuzzy implication algebra.

Theorem 3.2

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

Proofs

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

\begin{array}{l} µ (1) = µ (a \Rightarrow a) \geq µ (a \Rightarrow a) \land µ (a) \\ \geq µ (1) \land µ (a) = µ (a) . \end{array}

Hence μ(1) ≥ μ(a), ∀ a ∈ A.□

Theovem 3.3

If a fuzzy subset μ in A is a fuzzy implication algebra, then the following holds: $F_{A 1} : µ (a \Rightarrow 1) \geq µ (a) .$

F_{A 2} : µ (a \Rightarrow (b \Rightarrow 1)) = µ (1), \forall a, b \in A .

Proof:

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

1. 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.
2. 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.□

Definition 3.4

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].

Theorem 3.5

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.

Proof:

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.□

Theorem 3.6

If a fuzzy subset μ in A satisfy F_A1 and F_A2, then μ is a fuzzy implication algebra.

Definition 3.7

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.

Example 3.2

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.

Theorem 3.8

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

Proof.

Let μ be a fuzzy normal subset of A, and let a, b ∈ A. Then,

\begin{matrix} µ (a \Rightarrow b) = µ (1 \Rightarrow (a \Rightarrow b)) \\ = µ ((b \Rightarrow b) \Rightarrow (a \Rightarrow b)) \\ \geq µ (b \Rightarrow a) \land µ (b \Rightarrow b) \\ = µ (b \Rightarrow a) \land µ (1) = µ (b \Rightarrow a) \end{matrix}

Hence $µ (a \Rightarrow b) \geq µ (b \Rightarrow a), \forall a, b \in A \dots (1)$

Again, $µ (b \Rightarrow a) = µ (1 \Rightarrow (b \Rightarrow a))$

\geq µ (a \Rightarrow b) \land µ (a \Rightarrow a) = µ (a \Rightarrow b) \land µ (1) = µ (a \Rightarrow b) .

Hence $µ (b \Rightarrow a) \geq µ (a \Rightarrow b), \forall a, b \in A (2)$

Therefore $µ (a \Rightarrow b) = µ (b \Rightarrow a), \forall a, b \in A .$ □

= µ ((a \Rightarrow a) \Rightarrow (b \Rightarrow a))

Theorem 3.9

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.

Prof.

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

\begin{matrix} µ ((x \Rightarrow a) \Rightarrow (y \Rightarrow b)) \\ \geq µ (x \Rightarrow y) \land µ (a \Rightarrow b) \\ = µ (1) \land µ (1) = µ (1) \end{matrix}

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.□

Theorem 3.10

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

Proof.

Let {μ_α|α ∈ Λ} be a family of fuzzy normal implication algebra, and let a, b, x, y ∈ A. Then $\begin{matrix} (\cap_{α \in Λ} µ_{α}) ((x \Rightarrow a) \Rightarrow (y \Rightarrow b)) \\ = {Inf}_{α \in Λ} µ_{α} ((x \Rightarrow a) \Rightarrow (y \Rightarrow b)) \\ \geq {inf}_{α \in Λ} {min {µ_{α} (x \Rightarrow y), µ_{α} (a \Rightarrow b)}} \\ = min {{inf}_{α \in Λ} µ_{α} (x \Rightarrow y), {inf}_{α \in Λ} µ_{α} (a \Rightarrow b)} \\ = min {(\cap_{α \in Λ} µ_{α}) (x \Rightarrow y), (\cap_{α \in Λ}) µ_{α} (a \Rightarrow b)} \end{matrix}$

Hence ∩_α∈Λμ_α is a fuzzy normal set in A. Consequently, ∩_α∈Λμ_α is a fuzzy normal implication algebra.

Definition 3.11

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

1. μ(0) ≥ μ(a), ∀ a ∈ A.
2. μ(a) ≥ μ(a ⇒ b) ∧ μ(b), ∀ a, b ∈ A.

Example 3.3

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

1. μ(0) ≥ μ(a), ∀ a ∈ A|{0}.
2. μ(a ⇒ b) ≥ μ(a ⇒ b) ∧ μ(b), ∀ a, b ∈ A. Hence μ is a fuzzy implication ideal of A.

Table 3. 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

Theorem 3.12

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].

Proof:

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. Leta, 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.

Definition 3.13

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

1. μ(1) ≥ μ(a), ∀, a ∈ A.
2. μ(a ⇒ b) ≥ μ(a) ∧ μ(a ⇒ b), ∀ a, b ∈ A.

Example 3.4

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

1. μ(1) ≥ μ(a), ∀, a, b ∈ A|{1}.
2. μ(a ⇒ b) ≥ μ(a) ∧ μ(a ⇒ b), ∀ a, b ∈ A. Hence μ is a fuzzy implication filter of A.

Theorem 3.14

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.

Theorem 3.15

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

Table 4. 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.

References

1. Abbott JC: orthoimplication algebras. Stud. Logica. 1976; 35: 173–177. Publisher Full Text
2. Berhanu, et al.; B-Almost distributive Fuzzy Lattice. Bulletin of the section of Logic. 2018; Vol. 47(3): pp.171–185. Publisher Full Text
3. Tefera DG: Hilbert implication Algebra and some properties. Journal of Applied Mathematics, Hindawi. 2021: 5. Publisher Full Text
4. Gerima T, Endris Y, Fasil G: Ideals and Filters on implication algebras. Advances in Mathematics:Scientific Journals. 2021; Vol.10(3): 1167–1174. ISSN : 1857-8365(printed);1877- 8435(electronic).
5. Kim CB, Kim HS: On BM algebras. Sci.Math.Japo. 2001; 63: 421–427.
6. Roh EH, Kim SY, Xu Y, et al.: Some operations on Lattice Implication Algbras. IJMMS. 2001; 27(1): 45–52. Publisher Full Text
7. Neggers J, Kim HS: On B-algebras. Mat. Vesn. 2002; 54: 21–29.
8. Bandaru RK, et al.: Fuzzy implicative filters of BE-algebras based on the theory of falling shadows. Ser.Math.Inform. 2018; 33(1): 031–040. Publisher Full Text
9. Xu Y: Lattice implication algebras. J. Southwest Jiaotong Univ. 1993; 1: 20–27.
10. Xu Y, Qin KY: Lattice H implication algebras and lattice implication algebra classes. J.Hebei mining Civil Eng. Ins. 1992; 3: 139–143.
11. Xu Y, Qin KY: On filters of implication algebras. J.Fuzzy Math. 1993; 1: 251–260.
12. Jun YB, Roh EH, Kim HS: on fuzzy B- algebras. Czechoslovak Mathematical Jurnal. 2002; 52(2): 375–384. Publisher Full Text
13. Jun YB: Fuzzy positive implicative and fuzzy associative filters of lattice implication al- gebras. Fuzzy Sets Syst. 2001; 121: 353–357.
14. Zadeh LA: Fuzzy sets, information and Control.1965; 8: 338–353.

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VERSION 1 PUBLISHED 12 May 2026

Author details Author details

¹ Mathematics, Woldia University, Woldia, Amhara, 400, Ethiopia
² Mathematics, Wollo University, Dessie, Amhara, Ethiopia

Derebew Nigussie Derso
Roles: Conceptualization, Software, Validation, Writing – Review & Editing

Gerima Tefera Dejen
Roles: Conceptualization, Data Curation, Software, Supervision, Validation, Writing – Original Draft Preparation, Writing – Review & Editing

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

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Published: 12 May 2026, 15:723

https://doi.org/10.12688/f1000research.180502.1

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© 2026 Derso DN and Dejen GT. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[1] 1. Abbott JC: orthoimplication algebras. Stud. Logica. 1976; 35: 173–177. Publisher Full Text

[2] 2. Berhanu, et al.; B-Almost distributive Fuzzy Lattice. Bulletin of the section of Logic. 2018; Vol. 47(3): pp.171–185. Publisher Full Text

[3] 3. Tefera DG: Hilbert implication Algebra and some properties. Journal of Applied Mathematics, Hindawi. 2021: 5. Publisher Full Text

[4] 4. Gerima T, Endris Y, Fasil G: Ideals and Filters on implication algebras. Advances in Mathematics:Scientific Journals. 2021; Vol.10(3): 1167–1174. ISSN : 1857-8365(printed);1877- 8435(electronic).

[5] 5. Kim CB, Kim HS: On BM algebras. Sci.Math.Japo. 2001; 63: 421–427.

[6] 6. Roh EH, Kim SY, Xu Y, et al.: Some operations on Lattice Implication Algbras. IJMMS. 2001; 27(1): 45–52. Publisher Full Text

[7] 7. Neggers J, Kim HS: On B-algebras. Mat. Vesn. 2002; 54: 21–29.

[8] 8. Bandaru RK, et al.: Fuzzy implicative filters of BE-algebras based on the theory of falling shadows. Ser.Math.Inform. 2018; 33(1): 031–040. Publisher Full Text

[9] 9. Xu Y: Lattice implication algebras. J. Southwest Jiaotong Univ. 1993; 1: 20–27.

[10] 10. Xu Y, Qin KY: Lattice H implication algebras and lattice implication algebra classes. J.Hebei mining Civil Eng. Ins. 1992; 3: 139–143.

[11] 11. Xu Y, Qin KY: On filters of implication algebras. J.Fuzzy Math. 1993; 1: 251–260.

[12] 12. Jun YB, Roh EH, Kim HS: on fuzzy B- algebras. Czechoslovak Mathematical Jurnal. 2002; 52(2): 375–384. Publisher Full Text

[13] 13. Jun YB: Fuzzy positive implicative and fuzzy associative filters of lattice implication al- gebras. Fuzzy Sets Syst. 2001; 121: 353–357.

[14] 14. Zadeh LA: Fuzzy sets, information and Control.1965; 8: 338–353.

Fuzzy Ideals and Fuzzy Filters on Implication Algebras

Abstract

Background

Method

Result

Conclusion

Keywords

1. Introduction

2. Methods and materials

Theorem 2.1

Table 1. Fuzzy implication algerbra.

Example 2.1

Table 2. Fuzzy normal subset.

Definition 2.2

Propositions 2.2

Definition 2.3

Definition 2.4

Definition 2.5

3. Main Results

3.1 Fuzzy implication algebra

DefiniJion 3.1

Example 3.1

Theorem 3.2

Proofs

Theovem 3.3

Proof:

Definition 3.4

Theorem 3.5

Proof:

Theorem 3.6

Definition 3.7

Example 3.2

Theorem 3.8

Proof.

Theorem 3.9

Prof.

Theorem 3.10

Proof.

Definition 3.11

Example 3.3

Table 3. Fuzzy ideal.

Theorem 3.12

Proof:

Definition 3.13

Example 3.4

Theorem 3.14

Theorem 3.15

Table 4. Fuzzy filters.

4. Conclusion

Data availability

Acknowledgments

References

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