Categorical Equivalence between $PMV_f$- product algebras and semi-low $f_u$-rings

TL;DR

This paper establishes a categorical equivalence between a specific subvariety of $PMV$-algebras called $PMV_f$-algebras and semi-low $f_u$-rings, extending the representation theory of $MV$-algebras using prime spectra.

Contribution

It introduces a categorical equivalence between $PMV_f$-algebras and semi-low $f_u$-rings, expanding the algebraic and categorical understanding of these structures.

Findings

Categorical equivalence between $PMV_f$-algebras and semi-low $f_u$-rings.

Characterization of semi-low $f_u$-rings associated to Boolean algebras.

Proof that the class of $PMV_f$-algebras is coextensive.

Abstract

An explicit categorical equivalence is defined between a proper subvariety of the class of $PMV$-algebras, as defined by Di Nola and Dvure$\check{c}$enskij, to be called $PMV_f$-algebras, and the category of semi-low $f_u$-rings. This categorical representation is done using the prime spectrum of the $MV$-algebras, through the equivalence between $MV$-algebras and $l_u$-groups established by Mundici, from the perspective of the Dubuc-Poveda approach, that extends the construction defined by Chang on chains. As a particular case, semi-low $f_u$-rings associated to Boolean algebras are characterized. Besides we show that class of $PMV_f$-algebras is coextensive.