Duality for finitely valued algebras

TL;DR

This paper extends the theory of natural dualities to infinite algebras, establishing a categorical duality for a class of finitely valued algebras including MV-algebras and positive MV-algebras.

Contribution

It provides a universal algebraic framework for dualities involving infinite algebras, broadening the scope beyond finite cases.

Findings

Established a categorical duality for algebras representable as finite-range L-valued functions.

Extended duality theory to infinite L, covering MV-algebras.

Identified conditions on L for the duality to hold.

Abstract

The theory of natural dualities provides a well-developed framework for studying Stone-like dualities induced by an algebra $\mathbf{L}$ which acts as a dualizing object when equipped with suitable topological and relational structure. The development of this theory has, however, largely remained restricted to the case where $\mathbf{L}$ is finite. Motivated by the desire to provide a universal algebraic formulation of the existing duality of Cignoli and Marra or locally weakly finite MV-algebras and to extend it to a corresponding class of positive MV-algebras, in this paper we investigate Stone-like dualities where the algebra $\mathbf{L}$ is allowed to be infinite. This requires restricting our attention from the whole prevariety generated by $\mathbf{L}$ to the subclass of algebras representable as algebras of $\mathbf{L}$-valued functions of finite range, a distinction that does not arise in the case of finite $\mathbf{L}$. Provided some requirements on $\mathbf{L}$ are met, our main result establishes a categorical duality for this class of algebras, which covers the above cases of MV-algebras and positive MV-algebras.