On finite MTL-algebras that are representable as poset products of archimedean chains

TL;DR

This paper establishes a duality between finite MTL-algebras and finite labeled trees, characterizes certain poset products as sheaves over Alexandrov spaces, and provides explicit descriptions using direct products and ordinal sums.

Contribution

It introduces a duality framework and concrete descriptions for finite MTL-algebras constructed as poset products, advancing the structural understanding of these algebras.

Findings

Duality between finite MTL-algebras and finite labeled trees

Poset products of MTL-algebras as sheaves over Alexandrov spaces

Explicit descriptions via direct products and ordinal sums

Abstract

We obtain a duality between certain category of finite MTL-algebras and the category of finite labeled trees. In addition we prove that certain poset products of MTL-algebras are essentialy sheaves of MTL-chains over Alexandrov spaces. Finally we give a concrete description for the studied poset products in terms of direct products and ordinal sums of finite MTL-algebras.