Implicative Ideals of BCK-Algebras Based on the Fuzzy Sets and the Theory of Falling Shadows

Abstract

Based on the theory of falling shadows and fuzzy sets, the notion of a falling fuzzy implicative ideal of a BCK-algebra is introduced. Relations among falling fuzzy ideals, falling fuzzy implicative ideals, falling fuzzy positive implicative ideals, and falling fuzzy commutative ideals are given. Relations between fuzzy implicative ideals and falling fuzzy implicative ideals are provided.

1. Introduction and Preliminaries

1.1. Introduction

In the study of a unified treatment of uncertainty modelled by means of combining probability and fuzzy set theory, Goodman [1] pointed out the equivalence of a fuzzy set and a class of random sets. Wang and Sanchez [2] introduced the theory of falling shadows which directly relates probability concepts with the membership function of fuzzy sets. Falling shadow representation theory shows us the way of selection relaid on the joint degrees distributions. It is reasonable and convenient approach for the theoretical development and the practical applications of fuzzy sets and fuzzy logics. The mathematical structure of the theory of falling shadows is formulated in [3]. Tan et al. [4, 5] established a theoretical approach to define a fuzzy inference relation and fuzzy set operations based on the theory of falling shadows. Jun and Park [6] discussed the notion of a falling fuzzy subalgebra/ideal of a BCK/BCI-algebra. Jun and Kang [7, 8] also considered falling fuzzy positive implicative ideals and falling fuzzy commutative ideals. In this paper, we establish a theoretical approach to define a fuzzy implicative ideal in a BCK-algebra based on the theory of falling shadows. We consider relations between fuzzy implicative ideals and falling fuzzy implicative ideals. We provide relations among falling fuzzy ideals, falling fuzzy implicative ideals, falling fuzzy positive implicative ideals, and falling fuzzy commutative ideals.

1.2. Basic Results on BCK-Algebras and Fuzzy Aspects

A BCK/BCI-algebra is an important class of logical algebras introduced by Iséki and was extensively investigated by several researchers.

An algebra (𝑋;∗,0) of type (2,0) is called a BCI-algebra if it satisfies the following conditions: (i)(∀𝑥,𝑦,𝑧∈𝑋)(((𝑥∗𝑦)∗(𝑥∗𝑧))∗(𝑧∗𝑦)=0), (ii)(∀𝑥,𝑦∈𝑋)((𝑥∗(𝑥∗𝑦))∗𝑦=0), (iii)(∀𝑥∈𝑋)(𝑥∗𝑥=0), (iv)(∀𝑥,𝑦∈𝑋)(𝑥∗𝑦=0,𝑦∗𝑥=0⇒𝑥=𝑦). If a BCI-algebra 𝑋 satisfies the following identity: (v)(∀𝑥∈𝑋)(0∗𝑥=0), then 𝑋 is called a 𝐵𝐶𝐾-algebra. Any BCK-algebra 𝑋 satisfies the following axioms: (a1)(∀𝑥∈𝑋)(𝑥∗0=𝑥), (a2)(∀𝑥,𝑦,𝑧∈𝑋)(𝑥≤𝑦⇒𝑥∗𝑧≤𝑦∗𝑧,𝑧∗𝑦≤𝑧∗𝑥), (a3)(∀𝑥,𝑦,𝑧∈𝑋)((𝑥∗𝑦)∗𝑧=(𝑥∗𝑧)∗𝑦), where 𝑥≤𝑦 if and only if 𝑥∗𝑦=0.A subset 𝐼 of a BCK-algebra 𝑋 is called an ideal of 𝑋 if it satisfies the following: (b1)0∈𝐼, (b2)(∀𝑥∈𝑋)(∀𝑦∈𝐼)(𝑥∗𝑦∈𝐼⇒𝑥∈𝐼). Every ideal 𝐼 of a BCK-algebra 𝑋 has the following assertion: (∀𝑥∈𝑋)∀𝑦∈𝐼𝑥≤𝑦⟹𝑥∈𝐼. (1.1)A subset 𝐼 of a BCK-algebra 𝑋 is called a positive implicative ideal of 𝑋 if it satisfies (b1) and (b3)(∀𝑥,𝑦,𝑧∈𝑋)((𝑥∗𝑦)∗𝑧∈𝐼,𝑦∗𝑧∈𝐼⇒𝑥∗𝑧∈𝐼). A subset 𝐼 of a BCK-algebra 𝑋 is called a commutative ideal of 𝑋 if it satisfies (b1) and (b4)(∀𝑥,𝑦,𝑧∈𝑋)((𝑥∗𝑦)∗𝑧∈𝐼,𝑧∈𝐼⇒𝑥∗(𝑦∗(𝑦∗𝑥))∈𝐼). A subset 𝐼 of a BCK-algebra 𝑋 is called an implicative ideal of 𝑋 if it satisfies (b1) and (b5)(∀𝑥,𝑦,𝑧∈𝑋)((𝑥∗(𝑦∗𝑥))∗𝑧∈𝐼,𝑧∈𝐼⇒𝑥∈𝐼). We refer the reader to the paper [9] and book [10] for further information regarding BCK-algebras.A fuzzy set 𝜇 in a BCK-algebra 𝑋 is called a fuzzy ideal of 𝑋 (see [11]) if it satisfies the following: (c1)(∀𝑥∈𝑋)(𝜇(0)≥𝜇(𝑥)), (c2)(∀𝑥,𝑦∈𝑋)(𝜇(𝑥)≥min{𝜇(𝑥∗𝑦),𝜇(𝑦)}). A fuzzy set 𝜇 in a BCK-algebra 𝑋 is called a fuzzy positive implicative ideal of 𝑋 (see [12]) if it satisfies (c1) and (c3)(∀𝑥,𝑦,𝑧∈𝑋)(𝜇(𝑥∗𝑧)≥min{𝜇((𝑥∗𝑦)∗𝑧),𝜇(𝑦∗𝑧)}). A fuzzy set 𝜇 in a BCK-algebra 𝑋 is called a fuzzy commutative ideal of 𝑋 (see [13]) if it satisfies (c1) and (c4)(∀𝑥,𝑦,𝑧∈𝑋)(𝜇(𝑥∗(𝑦∗(𝑦∗𝑥)))≥min{𝜇((𝑥∗𝑦)∗𝑧),𝜇(𝑧)}). A fuzzy set 𝜇 in a BCK-algebra 𝑋 is called a fuzzy implicative ideal of 𝑋 (see [14]) if it satisfies (c1) and (c5)(∀𝑥,𝑦,𝑧∈𝑋)(𝜇(𝑥)≥min{𝜇((𝑥∗(𝑦∗𝑥))∗𝑧),𝜇(𝑧)}).

Proposition 1.1 (see [11, 14]). Let 𝜇 be a fuzzy set in a BCK-algebra 𝑋. Then 𝜇 is a fuzzy (implicative) ideal of 𝑋 if and only if (∀𝑡∈[0,1])𝜇_𝑡≠∅⟹𝜇_𝑡isanimplicativeidealof𝑋, (1.2) where 𝜇_𝑡∶={𝑥∈𝑋∣𝜇(𝑥)≥𝑡}.

1.3. The Theory of Falling Shadows

We first display the basic theory on falling shadows. We refer the reader to the papers [1–5] for further information regarding falling shadows.

Given a universe of discourse 𝑈, let 𝒫(𝑈) denote the power set of 𝑈. For each 𝑢∈𝑈, let 𝑢̇∶={𝐸∣𝑢∈𝐸,𝐸⊆𝑈}, (1.3) and for each 𝐸∈𝒫(𝑈), let 𝐸̇∶={𝑢̇∣𝑢∈𝐸}. (1.4) An ordered pair (𝒫(𝑈),ℬ) is said to be a hyper-measurable structure on 𝑈 if ℬ is a 𝜎-field in 𝒫(𝑈) and 𝑈̇⊆ℬ. Given a probability space (Ω,𝒜,𝑃) and a hyper-measurable structure (𝒫(𝑈),ℬ) on 𝑈, a random set on 𝑈 is defined to be a mapping 𝜉∶Ω→𝒫(𝑈) which is 𝒜-ℬ measurable, that is, (∀𝐶∈ℬ)𝜉⁻¹(𝐶)={𝜔∣𝜔∈Ω,𝜉(𝜔)∈𝐶}∈𝒜. (1.5) Suppose that 𝜉 is a random set on 𝑈. Let 𝐻(𝑢)∶=𝑃(𝜔∣𝑢∈𝜉(𝜔))foreach𝑢∈𝑈. (1.6) Then 𝐻 is a kind of fuzzy set in 𝑈. We call 𝐻 a falling shadow of the random set 𝜉, and 𝜉 is called a cloud of 𝐻.

For example, (Ω,𝒜,𝑃)=([0,1],𝒜,𝑚), where 𝒜 is a Borel field on [0,1] and 𝑚 is the usual Lebesgue measure. Let 𝐻 be a fuzzy set in 𝑈 and 𝐻_𝑡∶={𝑢∈𝑈∣𝐻(𝑢)≥𝑡} be a 𝑡-cut of 𝐻. Then 𝜉∶[0,1]⟶𝒫(𝑈),𝑡⟼𝐻_𝑡 (1.7) is a random set and 𝜉 is a cloud of 𝐻. We will call 𝜉 defined above as the cut-cloud of 𝐻 (see [1]).

2. Falling Fuzzy Implicative Ideals

In what follows let 𝑋 denote a BCK-algebra unless otherwise.

Definition 2.1 (see [6–8]). Let (Ω,𝒜,𝑃) be a probability space, and let 𝜉∶Ω⟶𝒫(𝑋) (2.1) be a random set. If 𝜉(𝜔) is an ideal (resp., positive implicative ideal and commutative ideal) of 𝑋 for any 𝜔∈Ω, then the falling shadow 𝐻 of the random set 𝜉, that is, 𝐻(𝑥)=𝑃(𝜔∣𝑥∈𝜉(𝜔)) (2.2) is called a falling fuzzy ideal (resp., falling fuzzy positive implicative ideal and falling fuzzy commutative ideal) of 𝑋.
Let (Ω,𝒜,𝑃) be a probability space and let 𝐹(𝑋)∶=𝑓∣𝑓∶Ω⟶𝑋isamapping, (2.3) where 𝑋 is a BCK-algebra. Define an operation ⊛ on 𝐹(𝑋) by (∀𝜔∈Ω)𝑓⊛𝑔(𝜔)=𝑓(𝜔)∗𝑔(𝜔), (2.4) for all 𝑓,𝑔∈𝐹(𝑋). Let 𝜃∈𝐹(𝑋) be defined by 𝜃(𝜔)=0 for all 𝜔∈Ω. Then (𝐹(𝑋);⊛,𝜃) is a BCK-algebra (see [6]).

Definition 2.2. Let (Ω,𝒜,𝑃) be a probability space and let 𝜉∶Ω⟶𝒫(𝑋) (2.5) be a random set. If 𝜉(𝜔) is an implicative ideal of 𝑋 for any 𝜔∈Ω, then the falling shadow 𝐻 of the random set 𝜉, that is, 𝐻(𝑥)=𝑃(𝜔∣𝑥∈𝜉(𝜔)) (2.6) is called a falling fuzzy implicative ideal of 𝑋.
For any subset 𝐴 of 𝑋 and 𝑓∈𝐹(𝑋), let 𝐴_𝑓(𝜔),(𝐹(𝑋(𝑋)(𝜔).∶=𝜔∈Ω∣𝑓∈𝐴𝜉∶Ω⟶𝒫)),𝜔⟼𝑓∈𝐹∣𝑓∈𝐴 (2.7) Then 𝐴_𝑓∈𝒜.

Theorem 2.3. If 𝐴 is an implicative ideal of 𝑋, then 𝜉(𝜔)=𝑓∈𝐹(𝑋)∣𝑓(𝜔)∈𝐴 (2.8) is an implicative ideal of 𝐹(𝑋).

Proof. Assume that 𝐴 is an implicative ideal of 𝑋 and let 𝜔∈Ω. Since 𝜃(𝜔)=0∈𝐴, we see that 𝜃∈𝜉(𝜔). Let 𝑓,𝑔,ℎ∈𝐹(𝑋) be such that (𝑓⊛(𝑔⊛𝑓))⊛ℎ∈𝜉(𝜔) and ℎ∈𝜉(𝜔). Then 𝑓(𝜔)∗𝑔(𝜔)∗𝑓(𝜔)∗ℎ(𝜔)=𝑓⊛𝑔⊛𝑓⊛ℎ(𝜔)∈𝐴 (2.9) and ℎ(𝜔)∈𝐴. Since 𝐴 is an implicative ideal of 𝑋, it follows from (b5) that 𝑓(𝜔)∈𝐴 and so 𝑓∈𝜉(𝜔). Hence 𝜉(𝜔) is an implicative ideal of 𝐹(𝑋).

Since 𝜉⁻¹𝑓̇=𝜔∈Ω∣𝑓∈𝜉(𝜔)=𝜔∈Ω∣𝑓(𝜔)∈𝐴=𝐴_𝑓∈𝒜, (2.10) we see that 𝜉 is a random set on 𝐹(𝑋). Let 𝐻𝑓=𝑃𝜔∣𝑓(𝜔)∈𝐴. (2.11) Then 𝐻 is a falling fuzzy implicative ideal of 𝐹(𝑋).

Example 2.4. Consider a BCK-algebra 𝑋={0,𝑎,𝑏,𝑐,𝑑} with a Cayley table which is given by Table 1 (see [10, page 274]). Let (Ω,𝒜,𝑃)=([0,1],𝒜,𝑚) and let 𝜉∶[0,1]→𝒫(𝑋) be defined by 𝜉(𝑡){}[),{}[),{}[),{}[].∶=0,𝑎if𝑡∈0,0.250,𝑏if𝑡∈0.25,0.550,𝑏,𝑑if𝑡∈0.55,0.70,𝑎,𝑏,𝑐if𝑡∈0.7,1 (2.12) Then 𝜉(𝑡) is an implicative ideal of 𝑋 for all 𝑡∈[0,1]. Hence 𝐻, which is given by 𝐻(𝑥)=𝑃(𝑡∣𝑥∈𝜉(𝑡)), is a falling fuzzy implicative ideal of 𝑋, and it is represented as follows: 𝐻(𝑥)=1if𝑥=0,0.55if𝑥=𝑎,0.75if𝑥=𝑏,0.3if𝑥=𝑐,0.15if𝑥=𝑑. (2.13) Then 𝐻_𝑡={0}(],{}(],{}(],{}(],[].if𝑡∈0.75,10,𝑎if𝑡∈0.55,0.750,𝑎,𝑏if𝑡∈0.3,0.550,𝑎,𝑏,𝑐if𝑡∈0.15,0.3𝑋if𝑡∈0,0.15 (2.14) If 𝑡∈(0.3,0.55], then 𝐻_𝑡={0,𝑎,𝑏} is not an implicative ideal of 𝑋 since (𝑐∗(𝑏∗𝑐))∗𝑎=(𝑐∗0)∗𝑎=𝑐∗𝑎=𝑏∈𝐻_𝑡 and 𝑎∈𝐻_𝑡, but 𝑐∉𝐻_𝑡. It follows from Proposition 1.1 that 𝐻 is not a fuzzy implicative ideal of 𝑋.

Table 1 

Cayley table.

Theorem 2.5. Every fuzzy implicative ideal of 𝑋 is a falling fuzzy implicative ideal of 𝑋.

Proof. Consider the probability space (Ω,𝒜,𝑃)=([0,1],𝒜,𝑚), where 𝒜 is a Borel field on [0,1] and 𝑚 is the usual Lebesque measure. Let 𝜇 be a fuzzy implicative ideal of 𝑋. Then 𝜇_𝑡 is an implicative ideal of 𝑋 for all 𝑡∈[0,1] by Proposition 1.1. Let 𝜉∶[0,1]⟶𝒫(𝑋) (2.15) be a random set and 𝜉(𝑡)=𝜇_𝑡 for every 𝑡∈[0,1]. Then 𝜇 is a falling fuzzy implicative ideal of 𝑋.

Example 2.4 shows that the converse of Theorem 2.5 is not valid.

Theorem 2.6. Every falling fuzzy implicative ideal is a falling fuzzy ideal.

Proof. Let 𝐻 be a falling fuzzy implicative ideal of 𝑋. Then 𝜉(𝜔) is an implicative ideal of 𝑋, and hence it is an ideal of 𝑋. Thus 𝐻 is a falling fuzzy ideal of 𝑋.

The converse of Theorem 2.6 is not true in general as shown by the following example.

Example 2.7. Consider a BCK-algebra 𝑋={0,𝑎,𝑏,𝑐,𝑑} with a Cayley table which is given by Table 2 (see [10, page 260]). Let (Ω,𝒜,𝑃)=([0,1],𝒜,𝑚) and let 𝜉∶[0,1]→𝒫(𝑋) be defined by 𝜉(𝑡){}[),{}[].∶=0,𝑐if𝑡∈0,0.30,𝑎,𝑏,𝑑if𝑡∈0.3,1 (2.16) Then 𝜉(𝑡) is an ideal of 𝑋 for all 𝑡∈[0,1]. Hence 𝐻(𝑥)=𝑃(𝑡∣𝑥∈𝜉(𝑡)) is a falling fuzzy ideal of 𝑋, and 𝐻(𝑥)={},0.3if𝑥=𝑐,0.7if𝑥∈𝑎,𝑏,𝑑1if𝑥=0. (2.17) In this case, we can easily check that 𝐻 is a fuzzy ideal of 𝑋 (see [6]). If 𝑡∈[0,0.3), then 𝜉(𝑡)={0,𝑐} is not an implicative ideal of 𝑋 since (𝑎∗(𝑏∗𝑎))∗𝑐∈𝜉(𝑡) and 𝑐∈𝜉(𝑡), but 𝑎∉𝜉(𝑡). Therefore 𝐻 is not a falling fuzzy implicative ideal of 𝑋.

Table 2 

Cayley table.

Theorem 2.8. Every falling fuzzy implicative ideal is both a falling fuzzy positive implicative ideal and a falling fuzzy commutative ideal.

Proof. Let 𝐻 be a falling fuzzy implicative ideal of 𝑋. Then 𝜉(𝜔) is an implicative ideal of 𝑋, and hence it is both a positive implicative ideal and a commutative ideal of 𝑋. Thus 𝐻 is both a falling fuzzy positive implicative ideal and a falling fuzzy commutative ideal.

The following example shows that the converse of Theorem 2.8 may not be true.

Example 2.9. (1) Consider a BCK-algebra 𝑋={0,𝑎,𝑏,𝑐,𝑑} with a Cayley table which is given by Table 3 (see [10, page 263]). Let (Ω,𝒜,𝑃)=([0,1],𝒜,𝑚) and let 𝜉∶[0,1]→𝒫(𝑋) be defined by 𝜉(𝑡){}[),{}[),{}[),{}[].∶=0,𝑐if𝑡∈0,0.150,𝑑if𝑡∈0.15,0.450,𝑎,𝑏if𝑡∈0.45,0.750,𝑐,𝑑if𝑡∈0.75,1 (2.18) Then 𝜉(𝑡) is a commutative ideal of 𝑋 for all 𝑡∈[0,1]. Hence 𝐻, which is given by 𝐻(𝑥)=𝑃(𝑡∣𝑥∈𝜉(𝑡)), is a falling fuzzy commutative ideal of 𝑋 (see [8]). But it is not a falling fuzzy implicative ideal of 𝑋 because if 𝑡∈[0.75,1] then 𝜉(𝑡)={0,𝑐,𝑑} is not an implicative ideal of 𝑋 since (𝑎∗(𝑏∗𝑎))∗𝑐=(𝑎∗𝑎)∗𝑐=0∗𝑐=0∈𝜉(𝑡) and 𝑐∈𝜉(𝑡), but 𝑎∉𝜉(𝑡).
(2) Let 𝑋={0,𝑎,𝑏,𝑐} be a BCK-algebra in which the ∗-multiplication is defined by Table 4. Let (Ω,𝒜,𝑃)=([0,1],𝒜,𝑚) and let [](𝑋){}[),{}[),{}[),[].𝜉∶0,1⟶𝒫,𝑡⟼0,𝑎if𝑡∈0,0.20,𝑏if𝑡∈0.2,0.50,𝑎,𝑏if𝑡∈0.5,0.9𝑋if𝑡∈0.9,1 (2.19) Then 𝜉(𝑡) is a positive implicative ideal of 𝑋 for all 𝑡∈[0,1]. Thus 𝐻, which is given by 𝐻(𝑥)=𝑃(𝑡∣𝑥∈𝜉(𝑡)), is a falling fuzzy positive implicative ideal of 𝑋. But it is not a falling fuzzy implicative ideal of 𝑋 because if 𝑡∈[0.2,0.5) then 𝜉(𝑡)={0,𝑏} is not an implicative ideal of 𝑋 since (𝑎∗(𝑐∗𝑎))∗𝑏=(𝑎∗𝑐)∗𝑏=0∗𝑏=0∈𝜉(𝑡) and 𝑏∈𝜉(𝑡), but 𝑎∉𝜉(𝑡).
The notions of a falling fuzzy positive implicative ideal and a falling fuzzy commutative ideal are independent, that is, a falling fuzzy commutative ideal need not be a falling fuzzy positive implicative ideal, and vice versa. In fact, the falling fuzzy commutative ideal 𝐻 in Example 2.9(1) is not a falling fuzzy positive implicative ideal. Also the falling fuzzy positive implicative ideal 𝐻 in Example 2.9(2) is not a falling fuzzy commutative ideal.
Let (Ω,𝒜,𝑃) be a probability space and 𝐻 a falling shadow of a random set 𝜉∶Ω→𝒫(𝑋). For any 𝑥∈𝑋, let Ω(𝑥;𝜉)∶={𝜔∈Ω∣𝑥∈𝜉(𝜔)}. (2.20) Then Ω(𝑥;𝜉)∈𝒜.

Table 3 

Cayley table.

Table 4 

∗-Multiplication.

Proposition 2.10. If a falling shadow 𝐻 of a random set 𝜉∶Ω→𝒫(𝑋) is a falling fuzzy implicative ideal of 𝑋, then (1)(∀𝑥,𝑦,𝑧∈𝑋)(Ω((𝑥∗(𝑦∗𝑥))∗𝑧;𝜉)∩Ω(𝑧;𝜉)⊆Ω(𝑥;𝜉)), (2)(∀𝑥,𝑦,𝑧∈𝑋)(Ω(𝑥;𝜉)⊆Ω((𝑥∗(𝑦∗𝑥))∗𝑧;𝜉)).

Proof. (1) Let 𝜔∈Ω((𝑥∗(𝑦∗𝑥))∗𝑧;𝜉)∩Ω(𝑧;𝜉). Then (𝑥∗(𝑦∗𝑥))∗𝑧∈𝜉(𝜔) and 𝑧∈𝜉(𝜔). Since 𝜉(𝜔) is an implicative ideal of 𝑋, it follows from (b5) that 𝑥∈𝜉(𝜔) so that 𝜔∈Ω(𝑥;𝜉). Therefore Ω𝑥∗𝑦∗𝑥∗𝑧;𝜉∩Ω(𝑧;𝜉)⊆Ω(𝑥;𝜉), (2.21) for all 𝑥,𝑦,𝑧∈𝑋.
(2) If 𝜔∈Ω(𝑥;𝜉), then 𝑥∈𝜉(𝜔). Since =)∗𝑥∗𝑦∗𝑥∗𝑧∗𝑥=𝑥∗𝑦∗𝑥∗𝑥∗𝑧(𝑥∗𝑥𝑦∗𝑥∗𝑧=0∗𝑦∗𝑥∗𝑧=0∗𝑧=0, (2.22) we have ((𝑥∗(𝑦∗𝑥))∗𝑧)∗𝑥=0∈𝜉(𝜔). Since 𝜉(𝜔) is an implicative ideal and hence an ideal of 𝑋, it follows that (𝑥∗(𝑦∗𝑥))∗𝑧∈𝜉(𝜔) and so 𝜔∈Ω((𝑥∗(𝑦∗𝑥))∗𝑧;𝜉). Hence Ω(𝑥;𝜉)⊆Ω((𝑥∗(𝑦∗𝑥))∗𝑧;𝜉) for all 𝑥,𝑦,𝑧∈𝑋.

Theorem 2.11. If 𝐻 is a falling fuzzy implicative ideal of 𝑋, then ∀𝑥,𝑦,𝑧∈𝑋𝐻(𝑥)≥𝑇_𝑚𝐻𝑥∗𝑦∗𝑥∗𝑧,𝐻(𝑧), (2.23) where 𝑇_𝑚(𝑠,𝑡)=max{𝑠+𝑡−1,0} for any 𝑠,𝑡∈[0,1].

Proof. By Definition 2.2, 𝜉(𝜔) is an implicative ideal of 𝑋 for any 𝜔∈Ω. Hence 𝜔∈Ω∣𝑥∗𝑦∗𝑥∗𝑧∈𝜉(𝜔)∩{𝜔∈Ω∣𝑧∈𝜉(𝜔)}⊆{𝜔∈Ω∣𝑥∈𝜉(𝜔)}, (2.24) and thus 𝐻(𝑥)((𝜔(𝜔∩{(𝜔(𝜔)((𝜔(𝜔)(𝜔)≥𝐻+𝐻(𝑧)=𝑃𝜔∣𝑥∈𝜉))≥𝑃𝜔∣𝑥∗𝑦∗𝑥∗𝑧∈𝜉)𝜔∣𝑧∈𝜉)}≥𝑃𝜔∣𝑥∗𝑦∗𝑥∗𝑧∈𝜉+𝑃𝜔∣𝑧∈𝜉))−𝑃𝜔∣𝑥∗𝑦∗𝑥∗𝑧∈𝜉or𝑧∈𝜉𝑥∗𝑦∗𝑥∗𝑧−1. (2.25) Therefore 𝐻(𝑥)≥max𝐻𝑥∗𝑦∗𝑥∗𝑧+𝐻(𝑧)−1,0=𝑇_𝑚𝐻𝑥∗𝑦∗𝑥∗𝑧,𝐻(𝑧). (2.26) This completes the proof.

Theorem 2.11 means that every falling fuzzy implicative ideal of 𝑋 is a 𝑇_𝑚-fuzzy implicative ideal of 𝑋.

Acknowledgment

The authors wish to thank the anonymous reviewers for their valuable suggestions.