Defining Homomorphisms and Other Generalized Morphisms of Fuzzy Relations in Monoidal Fuzzy Logics by Means of BK-Products

TL;DR

This paper extends the concept of generalized morphisms of relations into Monoidal Fuzzy Logics using relational inequalities and BK-products, unifying various fuzzy logic systems including BL, intuitionistic, and linear logics.

Contribution

It introduces a framework for defining homomorphisms and related morphisms in Monoidal Fuzzy Logics via relational inequalities and BK-products, generalizing previous fuzzy relation theories.

Findings

Relational inequalities over BK-products are used to define morphisms in monoidal fuzzy logics.

The framework subsumes theories based on BL, intuitionistic, and linear logics.

Provides a unified approach to fuzzy relation morphisms across multiple logic systems.

Abstract

The present paper extends generalized morphisms of relations into the realm of Monoidal Fuzzy Logics by first proving and then using relational inequalities over pseudo-associative BK-products (compositions) of relations in these logics. In 1977 Bandler and Kohout introduced generalized homomorphism, proteromorphism, amphimorphism, forward and backward compatibility of relations, and non-associative and pseudo-associative products (compositions) of relations into crisp (non-fuzzy Boolean) theory of relations. This was generalized later by Kohout to relations based on fuzzy Basic Logic systems (BL) of H\'ajek and also for relational systems based on left-continuous t-norms. The present paper is based on monoidal logics, hence it subsumes as special cases the theories of generalized morphisms (etc.) based on the following systems of logics: BL systems (which include the well known Goedel, product logic systems; Lukasiewicz logic and its extension to MV-algebras related to quantum logics), intuitionistic logics and linear logics.