Weak implicative filters in quasi-ordered residuated systems

Daniel A Romano

doi:10.22199/issn.0717-6279-4332

Abstract

The concept of residuated relational systems ordered under a quasiorder relation was introduced in 2018 by S. Bonzio and I. Chajda as a structure A = 〈A, ·,→, 1, R〉, where (A, ·) is a commutative monoid with the identity 1 as the top element in this ordered monoid under a quasi-order R. The author introduced and analyzed the concepts of filters and implicative filters in this type of algebraic structures. In this article, the concept of weak implicative filters in a quasi-ordered residuated system is introduced as a continuation of previous researches. Also, some conditions for a filter of such system to be a weak implicative filter are listed.

Highlights

Let (A, ·, 1) be a commutative semigroup with the identity 1
Definition 2.1. ([2], Definition 2.1) A residuated relational system is a structure A = hA, ·, →, 1, Ri, where hA, ·, →, 1i is an algebra of type h2, 2, 0i and R is a binary relation on A and satisfying the following properties: (1) (A, ·, 1) is a commutative monoid; (2) (∀x ∈ A)((x, 1) ∈ R); (3) (∀x, y, z ∈ A)((x · y, z) ∈ R ⇐⇒ (x, y → z) ∈ R)
The basic properties for residuated relational systems are subsumed in the following Theorem 2.1 ([2], Proposition 2.1)

Summary

Introduction

Let (A, ·, 1) be a commutative semigroup with the identity 1. The concept of residuated relational systems ordered under a quasiorder relation was introduced in 2018 by S. D. Maddux already mentioned, the first definition of relation algebras appears in [3] (cited by [4], page 441). D. Maddux, the first definition of relation algebras appears in [3]. This paper continues the investigations of quasi-ordered residuated systems and of their filters which were started in article [5]. Implicative filters in quasi-ordered residuated systems have been introduced and analyzed in the article [6]. The concept of weak implicative filters in a quasi-ordered residuated system is introduced as a continuation of the previous researches. Some conditions for a filter of such system to be a weak implicative filter are listed

Concept of quasi-ordered residuated systems

Concepts of filters

The concept of weak implicative filters