Elsevier

Fuzzy Sets and Systems

Volume 442, 15 August 2022, Pages 76-91
Fuzzy Sets and Systems

On bounded residuated EQ-algebras

https://doi.org/10.1016/j.fss.2021.04.022Get rights and content

Abstract

An EQ-algebra has three basic binary operations (meet, multiplication and a fuzzy equality) and a top element. An EQ-algebra is a lattice-ordered EQ-algebra satisfying the substitution property of the join operation. In this article, we study bounded residuated EQ-algebras (BR-EQ-algebras for short). We introduce a subvariety RL-EQ-algebras of BR-EQ-algebras, and prove that the categories of RL-EQ-algebras and residuated lattices are categorical isomorphic. We also prove that RL-EQ-algebras are precisely the BR-EQ-algebras that can be reconstructed from residuated lattices. We further show the existence of a closure operator on the poset of all BR-EQ-algebras with the same lattice and multiplication reduct, the existence of the maximum element in the poset. Then we introduce filters in BR-EQ-algebras and give a lattice isomorphism between the filter lattice and the congruence lattice. Finally, we prove that the category of residuated lattices is isomorphic to a reflective subcategory of BR-EQ-algebras.

Introduction

It is well known that the algebraic structure of truth values of fuzzy logics is a residuated lattice. Various logical algebras that proposed as the semantical systems of non-classical logic, for example MV-algebras [2], [3], [4], BL-algebras [9], MTL-algebras [7], lattice implication algebras [17] and R0-algebras [15], are all special classes of residuated lattices. Residuated lattices have been generally accepted to be the most basic and important logical algebras nowadays. They play an unshakable role in the study of fuzzy logics.

Another important algebraic structure in fuzzy logic is EQ-algebra. EQ-algebra was introduced by V. Novák [12] in order to provide an algebraic counterpart for the truth values of higher-order fuzzy logic. From then on many mathematicians begin to study properties of EQ-algebras, for example the interesting works [5], [6], [14], [16]. Since the implications in EQ-algebras are derived by fuzzy equalities, EQ-algebras provide a possibility to develop fuzzy logics with the basic connective being a fuzzy equality instead of an implication.

Since these two algebraic structures occupy an important place in fuzzy logics, it is a crucial research direction to find the relationships between residuated lattices and EQ-algebras. Because EQ-algebras relax the tie between multiplication and residuation, they seem to be a generalization of residuated lattices. Indeed, a residuated lattice is proved to be a special EQ-algebra but not vise versa. There exist EQ-algebras with different fuzzy equalities but the same implications.

Note that a residuated lattice has a bounded lattice reduct and satisfies the adjointness property. They seem to be quite similar to a proper class of EQ-algebras, namely bounded residuated lattice-ordered EQ-algebras. In addition, an EQ-algebra is a lattice-ordered EQ-algebra satisfying the substitution axiom of the join operation:((xy)z)(wx)z(wy), which holds naturally in the structure of residuated lattices with the derived biresiduums as fuzzy equalities. Based on the above reasons, it seems to be reasonable to transfer our focus to the relationships between bounded residuated EQ-algebras and residuated lattices. In this paper, we aim to investigate the connections between the categories of these two structures, and discuss the properties of bounded residuated EQ-algebras.

The paper is organized as follows. In Section 2, we review some basic definitions and properties of residuated lattices and EQ-algebras. In Section 3, we mainly discuss bounded residuated EQ-algebras (BR-EQ-algebras for short) and investigate the relationships with residuated lattices. We introduce RL-EQ-algebras, which is a subvariety of BR-EQ-algebras, and prove that the category of RL-EQ-algebras is categorical isomorphic to residuated lattices (cf. Theorem 3.3). Furthermore, there exists a unique RL-EQ-algebra in the poset of all BR-EQ-algebras with the same lattice and multiplication reduct. As a consequence, a linearly ordered BR-EQ-algebra must be an RL-EQ-algebra (cf. Corollary 3.11). In Section 4, we introduce (prime, maximal, RL-)filters in BR-EQ-algebras and prove that there is a lattice isomorphism between the filter lattice and the congruence lattice (cf. Theorem 4.9). Finally, the category of residuated lattices is shown to be isomorphic to a reflective subcategory of BR-EQ-algebras (cf. Theorem 4.25).

We list categories that will be used in this paper as follows:

Section snippets

Preliminary

For the convenience of the reader, we recall some necessary notions and results of residuated lattices and EQ-algebras in this section.

RL-EQ-algebras and residuated lattices

Throughout this paper, E=(E;,,,,0,1) is a bounded residuated EQ-algebra (BR-ℓEQ-algebra for short) unless specified otherwise. For all concepts related to categories, the reader is referred to [11].

If a BR-EQ-algebra E=(E;,,,,0,1) satisfiesxy=xy for any x,yE, we say that E is an RL-EQ-algebra.

“RL” here is the abbreviation for the short term “residuated lattice”. As we will prove, the category of residuated lattices is categorical isomorphic to the category of RL-EQ-algebras.

Proposition 3.1

Let E=(

Filters and congruences of BR-EQ-algebras

In this section, we mainly discuss filters and congruences in BR-EQ-algebras.

At first, we claim that the class of all BR-EQ-algebras is indeed an equational class, thus a variety [1].

Theorem 4.1

The class of all BR-ℓEQ-algebras is a variety.

Proof

All of (E1)-(E8) and inequality (3) can be described by identities. The bounded lattice structure can also be captured by identities. We just need to show that the adjointness property can be replaced by identities.

Consider the following two identities:(x((x(xz))y

Conclusion

In this paper we mainly discuss BR-EQ-algebras. The category of BR-EQ-algebras properly contains a full subcategory, which is categorical isomorphic to the category of residuated lattices. So from the category perspective, BR-EQ-algebras and residuated lattices are different. But the variety of BR-EQ-algebras is indeed quite similar to residuated lattices. We can transfer the filter theory of residuated lattices to BR-EQ-algebras without a lot of changes. In addition, there is a residuated

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by the National Science Foundation of China (Grants No. 11771004, 62050175).

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