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

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 $2$.