A topological approach to MTL-algebras

TL;DR

This paper introduces a topological duality framework for a broad class of MTL-algebras, linking their algebraic structure to topological spaces via dual constructions, enhancing understanding and representation.

Contribution

It provides a dualized construction for MTL-algebras using ordered quadruples and extends the Priestley duality approach for these algebras.

Findings

Unified topological construction for MTL-algebras

New results on Priestley duals of MTL and GMTL-algebras

Explicit dual representations from Boolean and radical components

Abstract

We dualize a construction of Aguzzoli-Flaminio-Ugolini of a large class of MTL-algebras from ordered quadruples consisting of a Boolean algebra, a generalized MTL-algebra, and two maps parameterizing the connection between these pieces. Our dualized construction gives a uniform way of building the extended Priestley duals of MTL-algebras in this class from the Stone duals of their Boolean skeletons, the extended Priestley duals of their radicals, and a family of closure operators associating the two. In order to facilitate this work, we also offer some new results regarding the extended Priestley duals of MTL-algebras and GMTL-algebras.