On Fuzzy Positive Implicative Filters in 𝐵𝐸-Algebras

Abstract

We study several degrees in defining a fuzzy positive implicative filter, which is a generalization of a fuzzy filter in 𝐵𝐸-algebras.

1. Introduction

In [1], H. S. Kim and Y. H. Kim introduced the notion of a 𝐵𝐸-algebra. Ahn and So [2, 3] introduced the notion of ideals in 𝐵𝐸-algebras. Ahn et al. [4] fuzzified the concept of 𝐵𝐸-algebras and investigated some of their properties. Jun and Ahn [5] provided several degrees in defining a fuzzy implicative filter.

In this paper, we study several degrees in defining a fuzzy positive implicative filter, which is a generalization of a fuzzy filter in 𝐵𝐸-algebras.

2. Preliminaries

We recall some definitions and results discussed in [1–3].

An algebra (𝑋;∗,1) of type (2, 0) is called a 𝐵𝐸-algebra if(BE1)𝑥∗𝑥=1 for all 𝑥∈𝑋,(BE2)𝑥∗1=1 for all 𝑥∈𝑋,(BE3)1∗𝑥=𝑥 for all 𝑥∈𝑋,(BE4)𝑥∗(𝑦∗𝑧)=𝑦∗(𝑥∗𝑧), for all 𝑥,𝑦,𝑧∈𝑋 (exchange).

We introduce a relation “≤” on a 𝐵𝐸-algebra 𝑋 by 𝑥≤𝑦, if and only if 𝑥∗𝑦=1. A nonempty subset 𝑆 of a 𝐵𝐸-algebra 𝑋 is said to be a subalgebra of 𝑋, if it is closed under the operation “∗.” By noticing that 𝑥∗𝑥=1, for all 𝑥∈𝑋, it is clear that 1∈𝑆. A 𝐵𝐸-algebra (𝑋;∗,1) is said to be self-distributive, if 𝑥∗(𝑦∗𝑧)=(𝑥∗𝑦)∗(𝑥∗𝑧), for all 𝑥,𝑦,𝑧∈𝑋.

Definition 1 (see [1]). Let (𝑋;∗,1) be a 𝐵𝐸-algebra and let 𝐹 be a nonempty subset of 𝑋. Then, 𝐹 is called a filter of 𝑋 if (F1)1∈𝐹,(F2)𝑥∗𝑦∈𝐹 and 𝑥∈𝐹 imply 𝑦∈𝐹,for all 𝑥,𝑦,𝑧∈𝑋.
A nonempty subset 𝐹 of a 𝐵𝐸-algebra 𝑋 is called an implicative filter of 𝑋 if it satisfies (F1) and (F3)𝑥∗(𝑦∗𝑧)∈𝐹 and 𝑥∗𝑦∈𝐹 imply 𝑥∗𝑧∈𝐹, for all 𝑥,𝑦,𝑧∈𝑋.

Example 2 (see [1]). Let 𝑋∶={1,𝑎,𝑏,𝑐,𝑑,0} be a 𝐵𝐸-algebra with the following table: ∗1𝑎𝑏𝑐𝑑01𝑎1𝑎𝑏𝑐𝑑0𝑏11𝑎𝑐𝑐𝑑𝑐111𝑐𝑐𝑐𝑑1𝑎𝑏1𝑎𝑏011𝑎11𝑎111111 (1) Then 𝐹₁∶={1,𝑎,𝑏} is a filter of 𝑋, but 𝐹₂∶={1,𝑎} is not a filter of 𝑋, since 𝑎∗𝑏∈𝐹₂ and 𝑎∈𝐹₂, but 𝑏∉𝐹₂.

Proposition 3. Let (𝑋;∗,1) be a 𝐵𝐸-algebra and let 𝐹 be a filter of 𝑋. If 𝑥≤𝑦 and 𝑥∈𝐹, for any 𝑦∈𝑋, then 𝑦∈𝐹.

Proposition 4. Let (𝑋;∗,1) be a self-distributive 𝐵𝐸-algebra. Then, the following hold, for any 𝑥,𝑦,𝑧∈𝑋: (i)if 𝑥≤𝑦, then 𝑧∗𝑥≤𝑧∗𝑦 and 𝑦∗𝑧≤𝑥∗𝑧;(ii)𝑦∗𝑧≤(𝑧∗𝑥)∗(𝑦∗𝑥);(iii)𝑦∗𝑧≤(𝑥∗𝑦)∗(𝑥∗𝑧).

A 𝐵𝐸-algebra (𝑋;∗,1) is said to be transitive if it satisfies Proposition 4 (iii).

Definition 5 (see [5]). A fuzzy subset 𝜇 of a 𝐵𝐸-algebra 𝑋 is called a fuzzy filter of 𝑋, if it satisfies, for all 𝑥,𝑦∈𝑋, (d1)𝜇(1)≥𝜇(𝑥),(d2)𝜇(𝑥)≥min{𝜇(𝑦∗𝑥),𝜇(𝑦)}. A fuzzy subset 𝜇 of a 𝐵𝐸-algebra 𝑋 is called a fuzzy implicative filter of 𝑋 if it satisfies (d1) and (d3)𝜇(𝑥∗𝑧)≥min{𝜇(𝑥∗(𝑦∗𝑧)),𝜇(𝑥∗𝑦)}, for all 𝑥,𝑦,𝑧∈𝑋.

Definition 6 (see [5]). Let 𝐹 be a nonempty subset of a 𝐵𝐸-algebra 𝑋 which is not necessarily a filter of 𝑋. One says that a subset 𝐺 of 𝑋 is an enlarged filter of 𝑋 related to 𝐹, if it satisfies the following:(1)𝐹 is a subset of 𝐺,(2)1∈𝐺,(3)(∀𝑥,𝑦∈𝑋)(∀𝑥∈𝐹)(𝑥∗𝑦∈𝐹⇒𝑦∈𝐺).

3. Fuzzy Positive Implicative Filters of 𝐵𝐸-Algebras with Degrees in (0,1]

𝑋 denotes a 𝐵𝐸-algebra unless specified otherwise.

Definition 7. A nonempty subset 𝐹 of 𝑋 is called a positive implicative filter of a 𝐵𝐸-algebra 𝑋 if it satisfies (F1) and (F4)𝑥∗((𝑦∗𝑧)∗𝑦)∈𝐹 and 𝑥∈𝐹 imply 𝑦∈𝐹, for all 𝑥,𝑦,𝑧∈𝑋.
Note that every positive implicative filter of a 𝐵𝐸-algebra 𝑋 is a filter of 𝑋.

Example 8. (1) Let 𝑋∶={1,𝑎,𝑏,𝑐,𝑑} be a self-distributive 𝐵𝐸-algebra ([1]) with the following table: ∗1𝑎𝑏𝑐𝑑1𝑎1𝑎𝑏𝑐𝑑𝑏11𝑏𝑐𝑑𝑐1𝑎1𝑐𝑐𝑑11𝑏1𝑏11111 (2) Then {1,𝑏} is an implicative filter of 𝑋 but not a positive implicative filter of 𝑋, since 𝑏∗((𝑎∗𝑑)∗𝑎)=1,𝑏∈{1,𝑏} and 𝑎∉{1,𝑏}.
(2) Consider a 𝐵𝐸-algebra 𝑋∶={1,𝑎,𝑏,𝑐,𝑑,0} as in Example 2. Then {1} is a filter of 𝑋 but not an implicative filter of 𝑋, since 𝑑∗(𝑎∗0)=1∈{1},𝑑∗𝑎=1∈{1}, and 𝑑∗0=𝑎∉{1}. Also, it is not a positive implicative filter of 𝑋, since 1∗((𝑎∗𝑏)∗𝑎)=1∈{1},1∈{1} and 𝑎∉{1}.
(3) Let 𝑋∶={1,𝑎,𝑏,𝑐} be a self-distributive 𝐵𝐸-algebra ([5]) with the following table: ∗1𝑎𝑏𝑐1𝑎1𝑎𝑏𝑐𝑏11𝑏𝑐𝑐1𝑎1𝑐11𝑏1 (3) Then {1,𝑏} is an implicative filter of 𝑋 and {1,𝑎,𝑏} is a positive implicative filter of 𝑋.

Definition 9. A fuzzy subset 𝜇 of a 𝐵𝐸-algebra 𝑋 is called a fuzzy positive implicative filter of 𝑋, if it satisfies (d1) and (d4)𝜇(𝑦)≥min{𝜇(𝑥∗((𝑦∗𝑧)∗𝑦)),𝜇(𝑥)}, for all 𝑥,𝑦∈𝑋.

Definition 10. Let 𝐹 be a nonempty subset of a 𝐵𝐸-algebra 𝑋 which is not necessarily a positive implicative filter of 𝑋. One says that a subset 𝐺 of 𝑋 is an enlarged positive implicative filter of 𝑋 related to 𝐹, if it satisfies the following:(1)𝐹 is a subset of 𝐺,(2)1∈𝐺,(3)(∀𝑥,𝑦,𝑧∈𝑋)(∀𝑥∈𝐹)(𝑥∗((𝑦∗𝑧)∗𝑦)∈𝐹⇒𝑦∈𝐺).
Obviously, every positive implicative filter is an enlarged positive implicative filter of 𝑋 related to itself. Note that there exists an enlarged positive implicative filter of 𝑋 related to any nonempty subset 𝐹 of 𝑋.

Example 11. Consider a 𝐵𝐸-algebra 𝑋={1,𝑎,𝑏,𝑐,𝑑} which is given in Example 8 (1). Note that 𝐹∶={1,𝑏} is not a positive implicative filter. Then, 𝐺∶={1,𝑎,𝑏,𝑐} is an enlarged positive implicative filter of 𝑋 related to 𝐹 and 𝐺 is not a positive implicative filter of 𝑋, since 𝑏∗((𝑑∗𝑑)∗𝑑)=𝑐∈𝐺,𝑏∈𝐺 and 𝑑∉𝐺.

Proposition 12. Let 𝐹 be a nonempty subset of a 𝐵𝐸-algebra 𝑋. Every enlarged positive implicative filter of 𝑋 related to 𝐹 is an enlarged filter of 𝑋 related to 𝐹.

Proof. Let 𝐺 be an enlarged positive implicative filter of 𝑋 related to 𝐹. By putting 𝑧∶=1 in Definition 10 (3), we have .∀𝑥,𝑦∈𝑋𝑥∗𝑦∗1∗𝑦=𝑥∗𝑦∈𝐹,𝑥∈𝐹⟹𝑦∈𝐺 (4) Hence, 𝐺 is an enlarged filter of 𝑋 related to 𝐹.

The converse of Proposition 12 is not true in general as seen in the following example.

Example 13. Let 𝑋∶={1,𝑎,𝑏,𝑐} be a transitive 𝐵𝐸-algebra ([2]) with the following table: ∗1𝑎𝑏𝑐1𝑎1𝑎𝑏𝑐𝑏11𝑎𝑎𝑐111𝑎11𝑎1 (5) Let 𝐹∶={1} and 𝐺∶={1,𝑐}. Then 𝐺 is an enlarged filter of 𝐹 but it is not an enlarged positive implicative filter of 𝐹, since 1∗((𝑎∗𝑐)∗𝑎)=1∈𝐹,1∈𝐹 and 𝑎∉𝐺.
In what follows let 𝜆 and 𝜅 be members of (0,1], and let 𝑛 and 𝑘 denote a natural number and a real number, respectively, such that 𝑘<𝑛, unless otherwise specified.

Definition 14 (see [5]). A fuzzy subset 𝜇 of a 𝐵𝐸-algebra 𝑋 is called a fuzzy filter of 𝑋 with degree (𝜆,𝜅), if it satisfies the following:(e1)(∀𝑥∈𝑋)(𝜇(1)≥𝜆𝜇(𝑥)),(e2)(∀𝑥,𝑦∈𝑋)(𝜇(𝑥)≥𝜅min{𝜇(𝑦∗𝑥),𝜇(𝑦)}).
A fuzzy subset 𝜇 of a 𝐵𝐸-algebra 𝑋 is called a fuzzy implicative filter of 𝑋 with degree (𝜆,𝜅), if it satisfies (e1) and(e3)(∀𝑥,𝑦,𝑧∈𝑋)(𝜇(𝑥∗𝑧)≥𝜅min{𝜇(𝑥∗(𝑦∗𝑧)),𝜇(𝑥∗𝑦)}).

Definition 15. A fuzzy subset 𝜇 of a 𝐵𝐸-algebra 𝑋 is called a fuzzy positive implicative filter of 𝑋 with degree (𝜆,𝜅), if it satisfies (e1) and(e4)(∀𝑥,𝑦,𝑧∈𝑋)(𝜇(𝑦)≥𝜅min{𝜇(𝑥∗((𝑦∗𝑧)∗𝑦)),𝜇(𝑥)}).

Proposition 16 (see [5]). Every fuzzy filter of a 𝐵𝐸-algebra 𝑋 with degree (𝜆,𝜅) satisfies the following assertions:(i)(∀𝑥,𝑦∈𝑋)(𝜇(𝑥∗𝑦)≥𝜆𝜅𝜇(𝑦)); (ii)(∀𝑥,𝑦∈𝑋)(𝑦≤𝑥⇒𝜇(𝑥)≥𝜆𝜅𝜇(𝑦)); (iii)(∀𝑥,𝑦,𝑧∈𝑋)(𝑥≤𝑦∗𝑧⇒𝜇(𝑧)≥min{𝜅𝜇(𝑦),𝜆𝜅²𝜇(𝑥)}).

Note that if 𝜆≠𝜅, then a fuzzy positive implicative filter with degree (𝜆,𝜅) may not be a fuzzy positive implicative filter with degree (𝜅,𝜆) and vice versa. Obviously, every fuzzy positive implicative filter is a fuzzy positive implicative filter with degree (𝜆,𝜅), but the converse may not be true.

Example 17. Consider a self-distributive 𝐵𝐸-algebra 𝑋={1,𝑎,𝑏,𝑐} which is given in Example 8 (3). Define a fuzzy subset 𝜇∶𝑋→[0,1] by 𝜇=.1𝑎𝑏𝑐0.70.40.80.4 (6) Then, 𝜇 is a fuzzy implicative filter of 𝑋 with degree (2/5,3/5) and a fuzzy filter of 𝑋 with degree (2/5,3/5), but it is neither a fuzzy filter of 𝑋 nor a fuzzy positive implicative filter of 𝑋 with degree (2/5,3/5) since 𝜇(1)(𝑏),𝜇(𝑎)3=0.7≱0.8=𝜇=0.4≱0.42=5=3×0.75(1)=3×𝜇5×min𝜇(1∗((𝑎∗𝑐)∗𝑎))=𝜇(1),𝜇(1). (7)

Example 18. Let 𝑋∶={1,𝑎,𝑏,𝑐} be a 𝐵𝐸-algebra ([2]) with the following table: ∗1𝑎𝑏𝑐1𝑎1𝑎𝑏𝑐𝑏11𝑏𝑐𝑐11111𝑎𝑐1 (8) Define a fuzzy subset 𝜇∶𝑋→[0,1] by 𝜇=.1𝑎𝑏𝑐0.70.80.40.4 (9) Then, 𝜇 is a fuzzy positive implicative filter of 𝑋 with degree (3/5,2/5). But it is neither a fuzzy filter of 𝑋 nor a fuzzy positive implicative filter of 𝑋 with degree (2/5,3/5) since 𝜇(1)(𝑎),𝜇(𝑐)3=0.7≱0.8=𝜇=0.4≱0.42=5=3×0.75(1)=3×𝜇5×min𝜇(𝑎∗((𝑐∗𝑏)∗𝑐))=𝜇(1),𝜇(𝑎). (10)
Also, it is not a fuzzy implicative filter of 𝑋 with degree (2/5,3/5) since 𝜇()(𝑐)3𝑐∗𝑏=𝜇=0.4≱0.42=5=3×0.75(1)=3×𝜇5𝜇(((1)()(1.×min𝑐∗𝑐∗𝑏))=𝜇,𝜇𝑐∗𝑐=𝜇) (11)

Proposition 19. If 𝜇 is a fuzzy positive implicative filter of a 𝐵𝐸-algebra 𝑋 with degree (𝜆,𝜅), then 𝜇 is a fuzzy filter of 𝑋 with degree (𝜆,𝜅).

Proof. By putting 𝑧∶=𝑦 in (e4), we have 𝜇𝑦𝜇(𝑥𝜇(𝑥≥𝜅min𝑥∗𝑦∗𝑦∗𝑦,𝜇)=𝜅min𝑥∗𝑦,𝜇) (12) for any 𝑥,𝑦∈𝑋. Thus, 𝜇 is a fuzzy filter of 𝑋 with degree (𝜆,𝜅).

The converse of Proposition 19 is not true in general (see Example 17).

Note that a fuzzy filter with degree (𝜆,𝜅) is a fuzzy filter if and only if (𝜆,𝜅)=(1,1).

Proposition 20. Let 𝜇 be a fuzzy positive implicative filter of a 𝐵𝐸-algebra 𝑋 with degree (𝜆,𝜅). Then, the following holds: ∀𝑥,𝑦∈𝑋𝜇(𝑥)≥𝜅𝜆𝜇𝑥∗𝑦∗𝑥. (13)

Proof. Assume that 𝜇 is a fuzzy positive implicative filter of a 𝐵𝐸-algebra 𝑋 with degree (𝜆,𝜅) and let 𝑥,𝑦∈𝑋. Using (e4) and (e1), we have 𝜇(𝑥)𝜇(1𝜇.≥𝜅min1∗𝑥∗𝑦∗𝑥,𝜇)=𝜅min𝑥∗𝑦∗𝑥,𝜆𝜇𝑥∗𝑦∗𝑥=𝜅𝜆𝜇𝑥∗𝑦∗𝑥 (14) This completes the proof.

Proposition 21. Let 𝜇 be a fuzzy filter of a 𝐵𝐸-algebra 𝑋 with degree (𝜆,𝜅) satisfying ∀𝑥,𝑦∈𝑋𝜇(𝑥)≥𝜇𝑥∗𝑦∗𝑥. (15) Then, 𝜇 is a positive implicative filter of 𝑋 with degree (𝜆,𝜅).

Proof. Let 𝑥,𝑦,𝑧∈𝑋. Using (e2), we have 𝜇𝑦𝜇(𝑥.≥𝜇𝑦∗𝑧∗𝑦≥𝜅min𝑥∗𝑦∗𝑧∗𝑦,𝜇) (16)

Thus, 𝜇 is a positive implicative filter of a 𝐵𝐸-algebra 𝑋 with degree (𝜆,𝜅).

Corollary 22. Let 𝜇 be a fuzzy filter of 𝑋. Then, 𝜇 is a fuzzy positive implicative filter of 𝑋, if and only if ∀𝑥,𝑦∈𝑋𝜇(𝑥)≥𝜇𝑥∗𝑦∗𝑥. (17)

Proof. It follows from Propositions 20 and 21.

Proposition 23. Every fuzzy positive implicative filter of a 𝐵𝐸-algebra 𝑋 with degree (𝜆,𝜅) satisfies the following assertions:(i)(∀𝑥,𝑦∈𝑋)(𝜇(𝑥∗𝑦)≥𝜆𝜅𝜇(𝑦));(ii)(∀𝑥,𝑦∈𝑋)(𝑥≤𝑦⇒𝜇(𝑦)≥𝜆𝜅𝜇(𝑥)).

Proof. It follows from Propositions 16 and 19.

Corollary 24. Let 𝜇 be a fuzzy positive implicative filter of a 𝐵𝐸-algebra 𝑋 with degree (𝜆,𝜅). If 𝜆=𝜅, then(i)(∀𝑥,𝑦∈𝑋)(𝜇(𝑥∗𝑦)≥𝜆²𝜇(𝑦)),(ii)(∀𝑥,𝑦∈𝑋)(𝑥≤𝑦⇒𝜇(𝑦)≥𝜆²𝜇(𝑥)).

Proposition 25. Let 𝑋 be a self-distributive 𝐵𝐸-algebra 𝑋. Let 𝜇 be a fuzzy positive implicative filter of 𝑋 with degree (𝜆,𝜅). Then, ∀𝑥,𝑦∈𝑋𝜇𝑦∗𝑥∗𝑥≥𝜆²𝜅³𝜇𝑦. (18)

Proof. Let 𝑥,𝑦∈𝑋. Let 𝜇 be a fuzzy positive implicative filter of 𝑋 with degree (𝜆,𝜅). By Proposition 19, 𝜇 is a fuzzy filter of 𝑋 with degree (𝜆,𝜅). Since 𝑥≤(𝑦∗𝑥)∗𝑥, using Proposition 4 (i), we have ((𝑦∗𝑥)∗𝑥)∗𝑦≤𝑥∗𝑦. Hence, ∗=∗≤∗.𝑥∗𝑦∗𝑦≤𝑦∗𝑥𝑥∗𝑦∗𝑥𝑥∗𝑦𝑦∗𝑥∗𝑥𝑦∗𝑥∗𝑥∗𝑦𝑦∗𝑥∗𝑥 (19) Using Propositions 16 and 23, we have 𝜇𝑦∗𝑥∗𝑥≥min𝜅𝜇𝑦∗𝑥∗𝑥∗𝑦,𝜆𝜅²𝜇𝜅𝑥∗𝑦∗𝑦≥min²𝜆𝜇𝑦,𝜆𝜅²𝜅𝜆𝜇𝑦=𝜅²𝜆min𝜇𝑦,𝜆𝜅𝜇𝑦=𝜅²𝜆(𝜆𝜅)𝜇𝑦=𝜆²𝜅³𝜇𝑦. (20) This completes the proof.

Definition 26. ([6]) Let 𝑋 be a 𝐵𝐸-algebra. 𝑋 is said to be commutative if the following identity holds:(C)(𝑥∗𝑦)∗𝑦=(𝑦∗𝑥)∗𝑥; that is, 𝑥∨𝑦=𝑦∨𝑥, where 𝑥∨𝑦=(𝑦∗𝑥)∗𝑥, for all 𝑥,𝑦∈𝑋.

Theorem 27. Let 𝑋 be a commutative self-distributive 𝐵𝐸-algebra. Every fuzzy positive implicative filter of 𝑋 with degree (𝜆,𝜅) is a fuzzy implicative filter of 𝑋 with degree (𝜆,𝜅³𝜆²).

Proof. Let 𝜇 be a fuzzy positive implicative filter of 𝑋 with degree (𝜆,𝜅). By Proposition 19, 𝜇 is a fuzzy filter of 𝑋 with degree (𝜆,𝜅). Using (BE4) and Proposition 4 (iii), we obtain (𝑥∗(𝑦∗𝑧))∗((𝑥∗𝑦)∗(𝑥∗(𝑥∗𝑧)))=1, for any 𝑥,𝑦,𝑧∈𝑋. Hence, by Proposition 16 (iii), we have 𝜇(𝑥∗(𝑥∗𝑧))≥min{𝜅𝜇(𝑥∗(𝑦∗𝑧)),𝜆𝜅²𝜇(𝑥∗𝑦)}. On the other hand, using (BE4) and (C), we obtain ))∗())))(∗(((().((𝑥∗𝑧∗𝑧𝑥∗𝑧=𝑥∗(((𝑥∗𝑧∗𝑧∗𝑧=𝑥∗((𝑧∗𝑥∗𝑧))𝑥∗𝑧))=𝑥∗1∗𝑥∗𝑧))=𝑥∗𝑥∗𝑧 (21) Using Proposition 20, we have 𝜇()))∗(((𝑥∗𝑧≥𝜅𝜆𝜇(((𝑥∗𝑧∗𝑧𝑥∗𝑧))=𝜅𝜆𝜇𝑥∗𝑥∗𝑧))≥𝜅𝜆min𝜅𝜇𝑥∗𝑦∗𝑧,𝜆𝜅²𝜇𝑥∗𝑦=𝜅²𝜆min𝜇𝑥∗𝑦∗𝑧,𝜅𝜆𝜇𝑥∗𝑦≥𝜅³𝜆²min𝜇𝑥∗𝑦∗𝑧,𝜇𝑥∗𝑦. (22) This completes the proof

Denote by ℱ_PI(𝑋) the set of all positive implicative filters of a 𝐵𝐸-algebra 𝑋. Note that a fuzzy subset 𝜇 of a 𝐵𝐸-algebra 𝑋 is a fuzzy positive implicative filter of 𝑋, if and only if (∀𝑡∈[0,1])𝑈𝜇;𝑡∈ℱ_PI(𝑋)∪0/. (23) But we know that, for any fuzzy subset 𝜇 of a 𝐵𝐸-algebra 𝑋, there exist 𝜆,𝜅∈(0,1) and 𝑡∈[0,1] such that(1)𝜇 is a fuzzy positive implicative filter of 𝑋 with degree (𝜆,𝜅),(2)𝑈(𝜇;𝑡)∉ℱ_PI(𝑋)∪{0/}.

Example 28. Consider a 𝐵𝐸-algebra 𝑋={1,𝑎,𝑏,𝑐,𝑑} which is given in Example 8 (1). Define a fuzzy subset 𝜇∶𝑋→[0,1] by 𝜇=.1𝑎𝑏𝑐𝑑0.40.30.50.30.3 (24) If 𝑡∈(0.4,0.6], then 𝑈(𝜇;𝑡)={1,𝑏} is not a positive implicative filter of 𝑋, since 𝑏∗((𝑎∗𝑑)∗𝑎)=1,𝑏∈{1,𝑏}, and 𝑎∉{1,𝑏}. But 𝜇 is a fuzzy positive implicative filter of 𝑋 with degree (0.4,0.6).

Theorem 29. Let 𝜇 be a fuzzy subset of a 𝐵𝐸-algebra 𝑋. For any 𝑡∈[0,1] with 𝑡≤max{𝜆,𝜅}, if 𝑈(𝜇;𝑡) is an enlarged positive implicative filter of 𝑋 related to 𝑈(𝜇;𝑡/max{𝜆,𝜅}), then 𝜇 is a fuzzy positive implicative filter of 𝑋 with degree (𝜆,𝜅).

Proof. Assume that 𝜇(1)<𝑡≤𝜆𝜇(𝑥), for some 𝑥∈𝑋 and 𝑡∈(0,𝜆]. Then 𝜇(𝑥)≥𝑡/𝜆≥𝑡/max{𝜆,𝜅} and so 𝑥∈𝑈(𝜇;𝑡/max{𝜆,𝜅}); that is, 𝑈(𝜇;𝑡/max{𝜆,𝜅})≠0/. Since 𝑈(𝜇;𝑡) is an enlarged filter of 𝑋 related to 𝑈(𝜇;𝑡/max{𝜆,𝜅}), we have 1∈𝑈(𝜇;𝑡); that is, 𝜇(1)≥𝑡. This is a contradiction, and thus 𝜇(1)≥𝜆𝜇(𝑥), for all 𝑥∈𝑋.
Now suppose that there exist 𝑎,𝑏,𝑐∈𝑋 such that 𝜇(𝑏)<𝜅min{𝜇(𝑎∗((𝑏∗𝑐)∗𝑏)),𝜇(𝑎)}. If we take 𝑡∶=𝜅min{𝜇(𝑎∗((𝑏∗𝑐)∗𝑏)),𝜇(𝑎)}, then 𝑡∈(0,𝜅]⊆(0,max{𝜆,𝜅}]. Hence, 𝑎∗((𝑏∗𝑐)∗𝑏)∈𝑈(𝜇;𝑡/𝜅)⊆𝑈(𝜇;𝑡/max{𝜆,𝜅}) and 𝑎∈𝑈(𝜇;𝑡/𝜅)⊆𝑈(𝜇;𝑡/max{𝜆,𝜅}). It follows from Definition 10 (3) that 𝑏∈𝑈(𝜇;𝑡) so that 𝜇(𝑏)≥𝑡, which is impossible. Therefore, 𝜇𝑦≥𝜅min𝜇𝑥∗𝑦∗𝑧∗𝑦,𝜇(𝑥) (25) for all 𝑥,𝑦,𝑧∈𝑋. Thus, 𝜇 is a fuzzy positive implicative filter of 𝑋 with degree (𝜆,𝜅).

Corollary 30. Let 𝜇 be a fuzzy subset of a 𝐵𝐸-algebra 𝑋. For any 𝑡∈[0,1] with 𝑡≤𝑘/𝑛, if 𝑈(𝜇;𝑡) is an enlarged positive implicative filter of 𝑋 related to 𝑈(𝜇;(𝑛/𝑘)𝑡), then 𝜇 is a fuzzy positive implicative filter of 𝑋 with degree (𝑘/𝑛,𝑘/𝑛).

Theorem 31. Let 𝑡∈[0,1] be such that 𝑈(𝜇;𝑡)(≠0/) is not necessarily a positive implicative filter of a 𝐵𝐸-algebra 𝑋. If 𝜇 is a fuzzy positive implicative filter of 𝑋 with degree (𝜆,𝜅), then 𝑈(𝜇;𝑡min{𝜆,𝜅}) is an enlarged positive implicative filter of 𝑋 related to 𝑈(𝜇;𝑡).

Proof. Since 𝑡min{𝜆,𝜅}≤𝑡, we have 𝑈(𝜇;𝑡)⊆𝑈(𝜇;𝑡min{𝜆,𝜅}). Since 𝑈(𝜇;𝑡)≠0/, there exists 𝑥∈𝑈(𝜇;𝑡) and so 𝜇(𝑥)≥𝑡. By (e1), we obtain 𝜇(1)≥𝜆𝜇(𝑥)≥𝜆𝑡≥𝑡min{𝜆,𝜅}. Therefore, 1∈𝑈(𝜇;𝑡min{𝜆,𝜅}).
Let 𝑥,𝑦,𝑧∈𝑋 be such that 𝑥∗((𝑦∗𝑧)∗𝑦)∈𝑈(𝜇;𝑡) and 𝑥∈𝑈(𝜇;𝑡). Then 𝜇(𝑥∗((𝑦∗𝑧)∗𝑦))≥𝑡 and 𝜇(𝑥)≥𝑡. It follows from (e4) that 𝜇𝑦𝜇(𝑥{};≥𝜅min𝑥∗𝑦∗𝑧∗𝑦,𝜇)≥𝜅𝑡≥𝑡min𝜆,𝜅 (26) so that 𝑦∈𝑈(𝜇;𝑡min{𝜆,𝜅}). Thus, 𝑈(𝜇;𝑡min{𝜆,𝜅}) is an enlarged positive implicative filter of 𝑋 related to 𝑈(𝜇;𝑡).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful to the referee for their valuable suggestions and help.