Abstract

Some new properties of fuzzy associative filters (also known as fuzzy associative pseudo-filters), fuzzy p-filter (also known as fuzzy pseudo-p-filters), and fuzzy a-filter (also known as fuzzy pseudo-a-filters) in pseudo-BCI algebras are investigated. By these properties, the following important results are proved: (1) a fuzzy filter (also known as fuzzy pseudo-filters) of a pseudo-BCI algebra is a fuzzy associative filter if and only if it is a fuzzy a-filter; (2) a filter (also known as pseudo-filter) of a pseudo-BCI algebra is associative if and only if it is an a-filter (also call it pseudo-a filter); (3) a fuzzy filter of a pseudo-BCI algebra is fuzzy a-filter if and only if it is both a fuzzy p-filter and a fuzzy q-filter.

Highlights

  • Introduction and PreliminariesIn the field of artificial intelligence research, nonclassical logics are extensively used

  • In this paper we discuss a kind of logic algebra system, that is, pseudoBCI algebra, which originated from BCI-logic; it is a kind of nonclassical logic and inspired by the calculus of combinators [2]

  • In [3], we show that pseudo-BCI algebra plays an important role in weakly integral residuated algebraic structure, which is in close connection with various fuzzy logic formal systems [4, 5]

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Summary

Introduction

Introduction and PreliminariesIn the field of artificial intelligence research, nonclassical logics (fuzzy logic, epistemic logic, nonmonotonic logic, default logic, etc.) are extensively used (see [1]). A fuzzy set μ : X → [0, 1] is called a fuzzy pseudofilter (briefly, fuzzy filter) of pseudo-BCI algebra X if it satisfies (FF1) μ(1) ≥ μ(x), ∀x ∈ X; (FF2) μ(y) ≥ min{μ(x → y), μ(x)}, ∀x, y ∈ X; (FF3) μ(y) ≥ min{μ(x 󴁄󴀼 y), μ(x)}, ∀x, y ∈ X.

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