Maximal theories of product logic

TL;DR

This paper investigates the structure of maximal filters in product algebras, establishing their correspondence with product hoops and characterizing their equational theory within fuzzy logic.

Contribution

It provides a novel characterization of maximal filters in product algebras via their relation to product hoops, enhancing understanding of their algebraic structure.

Findings

Maximal filters of product algebras are exactly the product hoops.

Constructs a product algebra from any given product hoop.

Characterizes the equational theory of maximal filters.

Abstract

Product logic is one of the main fuzzy logics arising from a continuous t-norm, and its equivalent algebraic semantics is the variety of product algebras. In this contribution, we study maximal filters of product algebras, and their relation with product hoops. The latter constitute the variety of 0-free subreducts of product algebras. Given any product hoop, we construct a product algebra of which the product hoop is (isomorphic to) a maximal filter. This entails that product hoops coincide exactly with the maximal filters of product algebras, seen as residuated lattices. In this sense, we characterize the equational theory of maximal filters of product algebras.