TL-filters of integral residuated l-monoids

doi:10.1016/j.ins.2006.03.019

Information Sciences

Volume 177, Issue 3, 1 February 2007, Pages 887-896

https://doi.org/10.1016/j.ins.2006.03.019 Get rights and content

Abstract

In this paper, we introduce and discuss the concept of TL-filters of integral residuated l-monoids. First we study some basic properties of TL-filters and give the formula for calculating the TL-filters generated by L-subsets. Then we discuss some properties of TL-filters under homomorphisms and study the relationship between TL-filters and T-congruence L-relations on integral commutative residuated l-monoids.

Introduction

The integral commutative residuated l-monoid (i.e., residuated lattice) is an important class of logical algebras [5], and the typical example of integral commutative residuated l-monoid is the interval [0, 1] endowed with the structure induced by a left-continuous t-norm [5]. The well-known integral commutative residuated l-monoids have R₀-algebras [12], BL-algebras [4], MTL-algebras [3] and so on. The filter theory plays an important role in studying these logical algebras (see [3], [4], [5], [11], [12]) and ordered semigroups [7]. From a logical point of view, a filter corresponds to a set of provable formulas. Sometimes, a filter is also called a deductive system (see [11]).

It is meaningful to study the fuzzy structure of filters on logical algebras. Based on the fuzzy set theory, some authors introduced the concepts of fuzzy implicative filters of R₀-algebras [8], fuzzy filters of BL-algebras [9], [10], and fuzzy filters of MTL-algebras [6]. It is worth noting that R₀-algebras, BL-algebras and MTL-algebras are all integral residuated l-monoids. Thus it is natural to extend fuzzy concepts on these logical algebras to integral residuated l-monoids. Motivated by these works [6], [8], [9], [10], we fuzzify the concept of filters on an integral residuated l-monoid in this paper. Using a general infinitely ∨-distributive t-norm T on a complete Brouwerian lattice L, we introduce the concept of TL-filters of integral residuated l-monoids, give the formula for calculating the TL-filters generated by L-subsets, discuss some properties of TL-filters under homomorphisms, and study the relationships between TL-filters and T-congruence L-relations on integral commutative residuated l-monoids.

The lattice properties required in this paper can be found in Birkhoff [1]. The terminologies and notations about t-norms, L-subsets, L-relations are in agreement with the ones in [13], [14], [15].

Throughout this paper, unless otherwise stated, L always represents any given complete Brouwerian lattice with the maximal element 1 and the minimal element 0 (see [1]), T any given infinitely ∨-distributive t-norm on L (see [13], [15]), and $N$ the set of natural numbers. Besides this, for any set X and a ∈ L, we set a_X(x) = a if x ∈ X and a_X(x) = 0 otherwise.

Section snippets

Preliminaries

In this section, for the purpose of reference, we present some definitions and results about integral residuated l-monoids and T-equivalence L-relations.

Definition 2.1

[2], [5]. A triple (S, ⩽, ∗) is called an integral residuated l-monoid if and only if the following three conditions are satisfied:

(1)
(S, ⩽, ∨, ∧, 0, 1) is a lattice, where ∨, ∧, 0, 1, respectively, stand for the join operation on S, the meet operation on S, the bottom element of S and the top element of S, and 0 ≠ 1.
(2)
(S, ∗) is a monoid with the identity 1.
(3)

TL-filters

In this section, we fuzzify the concept of filters on an integral residuated l-monoid (S, ⩽, ∗), introduce the concepts of TL-filter of S and discuss some of their basic properties.

Definition 3.1

By a TL-filter of S, we mean an L-subset μ of S (i.e., μ ∈ L^S) that satisfies the following conditions:

(TLF1)
μ is order-preserving, that is, if x, y ∈ S and x ⩽ y, then μ(x) ⩽ μ(y),
(TLF2)
μ(x ∗ y) ⩾ μ(x)Tμ(y) ∀ x, y ∈ S,
(TLF3)
μ(1) = 1.

In particular, a TL-filter of S is simply called an L-filter of S when T = ∧. The set of all TL-filters of S and the set of

Generated TL-filters by L-subsets

Given μ ∈ L^S, it follows from Theorem 3.1 that ⋀ {ν ∈ TLF(S)∣ν ⩾ μ} is the smallest TL-filter of S containing μ. Here we call this TL-filter the TL-filter of S generated by μ, and denote it by 〈μ〉_T.

It is easy to verify that for any μ, ν ∈ L^S,

(1) 〈1_{1}〉_T =〈0_S〉_T = 1_{1},
(2) μ ⩽ ν ⇒ 〈μ〉_T ⩽ 〈ν〉_T,
(3) if μ ∈ TLF(S), then 〈μ〉_T = μ.

Below, we give the formula for calculating the TL-filters generated by L-subsets.

Theorem 4.1

Let μ ∈ L^S. Then 〈μ〉_T(1) = 1 and $〈 μ 〉_{T} (x) = ⋁ {μ (x_{1}) T \dots T μ (x_{n}) | x ⩾ x_{1} * \dots * x_{n}, x_{1}, \dots, x_{n} \in S, n \in N}$ when x ≠ 1.

Proof

Let ν(1) = 1 and $ν (x) = ⋁ {μ (x_{1}) T \dots T$

Homomorphisms

Let (H; ⩽, ∗) be also an integral residuated l-monoid and ϕ a mapping from S onto H. ϕ is called a homomorphism of S into H (see [5]) if and only if ϕ is a lattice-homomorphism and a monoid-homomorphism satisfying the following additional condition: $ϕ (x \to y) = ϕ (x) \to ϕ (y) \forall x, y \in S .$

If ϕ is a homomorphism of S onto H, then

(1) ϕ is isotonic, that is, if x ⩽ y, x, y ∈ S then ϕ(x) ⩽ ϕ(y);
(2) $ϕ^{- 1} (1) = {x \in S | ϕ (x) = 1} \in F$ .

Let ϕ be a mapping from S into H, μ ∈ L^S, and ν ∈ L^H. The L-subsets ϕ(μ) ∈ L^H and ϕ⁻¹(ν) ∈ L^S, defined by $ϕ (μ) (y) = ⋁$

T-congruence L-relations on integral commutative residuated l-monoids

Let (S, ⩽, ∗) be an integral commutative residuated l-monoid. Finally, we study the relationship between TL-filters and T-congruence L-relations on S.

Definition 6.1

Let P ∈ TEL(S). P is called a T-congruence L-relation on S if it satisfies

(C1) P(x ∗ z, y ∗ z) ⩾ P(x, y) ∀ x, y, z ∈ S,
(C2) P(x → z, y → z) ⩾ P(x, y), P(z → x, z → y) ⩾ P(x, y) ∀ x,y,z ∈ S.

In particular, by a congruence L-relation on S, we mean a T-congruence L-relation on S when T = ∧.

We denote by TCL(S) and CL(S), respectively, the set of all T-congruence L-relations and the set

Conclusions

Since Zadeh proposed the notion of fuzzy sets, his ideas have been applied to various fields. In this paper, we apply fuzzy sets to integral residuated l-monoids and introduce the concept of TL-filters. We study some basic properties of TL-filters, obtain the formula for calculating the TL-filters generated by L-subsets, discuss some properties of TL-filters under homomorphisms, and derive the relationship between TL-filters and T-congruence L-relations on integral residuated commutative l

Acknowledgements

This work was done during the author’s visit to Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences. The author wishes to thank Dr. Liming Ge for his help during the preparation of this paper, Dr. Witold Pedrycz, Editor-in-Chief of the journal and the anonymous referees for their valuable comments and suggestions.

References (16)

F. Esteva et al.
Monoidal t-norm based logic: towards a logic for left-continuous t-norms
Fuzzy Sets Syst.
(2001)
Y.B. Jun et al.
Fuzzy filters of MTL-algebras
Inform. Sci.
(2005)
N. Kehayopulu et al.
Fuzzy bi-ideals in ordered semigroups
Inform. Sci.
(2005)
L.Z. Liu et al.
Fuzzy filters of BL-algebras
Inform. Sci.
(2005)
Z.D. Wang
On L-subsets and TL-subalgebras
Fuzzy Sets Syst.
(1994)
Z.D. Wang et al.
On T-congruence L-relations on groups and rings
Fuzzy Sets Syst.
(2001)
Y.D. Yu et al.
TL-subrings and TL-ideals. Part 1: Basic concepts
Fuzzy Sets Syst.
(1994)
L.A. Zadeh
Similarity relations and fuzzy ordering
Inform. Sci.
(1971)

There are more references available in the full text version of this article.

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^☆: This project is supported by the Natural Science Foundation of Zhejiang Province Education Commission and Science Foundation of Zhejiang Wanli University.

¹: Present address: Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, People’s Republic of China.

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Information Sciences

Abstract

Introduction

Section snippets

Preliminaries

TL-filters

Generated TL-filters by L-subsets

Homomorphisms

T-congruence L-relations on integral commutative residuated l-monoids

Conclusions

Acknowledgements

References (16)

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Inform. Sci.

Fuzzy filters of BL-algebras

Inform. Sci.

On L-subsets and TL-subalgebras

Fuzzy Sets Syst.

On T-congruence L-relations on groups and rings

Fuzzy Sets Syst.

TL-subrings and TL-ideals. Part 1: Basic concepts

Fuzzy Sets Syst.

Similarity relations and fuzzy ordering

Inform. Sci.

Cited by (14)

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On the T-partial order and properties

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TL-filters of integral residuated l-monoids☆

Abstract

Introduction

Section snippets

Preliminaries

TL-filters

Generated TL-filters by L-subsets

Homomorphisms

T-congruence L-relations on integral commutative residuated l-monoids

Conclusions

Acknowledgements

Fuzzy Sets Syst.

Inform. Sci.

Inform. Sci.

Inform. Sci.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Inform. Sci.