The spectrum problem for Abelian l-groups and MV-algebras

TL;DR

This paper characterizes topological spaces that are prime spectra of MV-algebras and Abelian l-groups, linking spectral spaces to lattice structures and providing logical axiomatizations.

Contribution

It provides a characterization of spectra of MV-algebras and Abelian l-groups using spectral space properties and lattice conditions, extending previous understanding.

Findings

Spectral spaces characterize prime spectra of MV-algebras.

Lattice of compact open sets relates to cylinder rational polyhedra.

Axiomatization of lattice structures in monadic second order logic.

Abstract

This paper deals with the problem of characterizing those topological spaces which are homeomorphic to the prime spectra of MV-algebras or Abelian l-groups. As a first main result, we show that a topological space $X$ is the prime spectrum of an MV-algebra if and only if: (1) $X$ is spectral, and (2) the lattice of compact open subsets of $X$ is an epimorphic image of a lattice of "cylinder rational polyhedra" (a natural generalization of rational polyhedra) of some hypercube. As a second main result we extend our results to Abelian l-groups. That is, let $X$ be a spectral space and $K(X)$ the lattice of its compact open sets. The following are equivalent: (1) $X$ is the spectrum of some Abelian l-group; (2) $X$ is homeomorphic to $Spec(K(X))$ and $K(X)\cup\{\infty\}$ is isomorphic to the lattice of the compact open sets of a local MV-algebra, where $\infty>x$ for every $x\in K(X)$. Finally we axiomatize, in monadic second order logic, the lattices of cylinder rational polyhedra of dimension $1$ and $2$.