A general view of the algebraic semantics of \L ukasiewicz logic with product

TL;DR

This paper explores the algebraic semantics of extended Lukasiewicz logic by establishing connections between algebraic classes through category-theoretic methods, tensor products, and categorical equivalences, advancing the understanding of logical algebraic structures.

Contribution

It introduces a novel categorical framework linking algebraic classes for Lukasiewicz logic extensions using tensor products and adjunctions, and applies these to prove amalgamation and transfer results.

Findings

Established adjunctions between algebraic classes

Defined tensor PMV-algebra for semisimple MV-algebras

Proved amalgamation properties and categorical equivalences

Abstract

This paper aims at connecting the various classes that provide an algebraic semantics for three different conservative expansions of Lukasiewicz logic, using algebraic and category-theoretical techniques. We connect such classes of algebras by adjunctions, using the tensor product of MV-algebras and defining the tensor PMV-algebra of a semisimple MV-algebra, inspired by the construction of the tensor algebra of a vector space. We further apply the main results to prove amalgamation properties and, via categorical equivalence, we transfer all results to the framework of lattice- ordered groups.