The spectrum problem for Abelian $$\ell $$-groups and MV-algebras

This paper deals with the problem of characterizing those topological spaces which are homeomorphic to the prime spectra of MV-algebras or Abelian $$\ell $$ -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 X is spectral, and the lattice K(X) of compact open subsets of X is a closed epimorphic image of the lattice of “cylinder rational polyhedra” (a natural generalization of rational polyhedra) of $$[0,1]^Y$$ for some set Y. As a second main result we extend our results to Abelian $$\ell $$ -groups. That is, a topological space X is the prime spectrum of an Abelian $$\ell $$ -group if and only if X is generalized spectral, and the lattice K(X) is a closed epimorphic image of the lattice of “cylinder rational cones” (a generalization of rational cones) in $${{\mathbb {R}}}^Y$$ for some set Y. Finally, we axiomatize, in monadic second order logic, the Belluce lattices of free MV-algebras (equivalently, the lattice of cylinder rational polyhedra) of dimension 1, 2 and infinite, and we study the problem of describing Belluce lattices in certain fragments of second order logic.