Spectra of finitely presented lattice - ordered Abelian groups and MV - algebras , part

(See e.g. [6] for background.) When X is equipped with a distinguished basis D for its topology, closed under finite meets and joins, one can investigate situations where D is also closed under the implication (2), i.e., where D is a Heyting subalgebra of O (X). Recall that X is a spectral space if it is compact and T0, its collection D of compact open subsets forms a basis which is closed under finite intersections and unions, and X is sober : any closed set that cannot be written as the union of two proper closed subsets, has a dense point. (In this case, the latter point is unique, because X is T0.) By Stone duality, spectral spaces are precisely the spaces arising as sets of prime ideals of some distributive lattice, topologised with the Stone or hull-kernel topology. Specifically, given such a spectral space X, its collection of compact open sets D is (naturally isomorphic to) the distributive lattice dual to X under Stone duality. We are going to exhibit a significant class of such spaces for which D is a Heyting subalgebra of O (X). We work with lattice-ordered Abelian groups and vector spaces. Using Mundici’s Γ-functor [8] the results can be rephrased in terms of MV-algebras, the algebraic semantics of Lukasiewicz infinite-valued propositional logic [5]. Recall that a lattice-ordered Abelian group [4], or `-group for short, is an Abelian group which is also a lattice, and is such that the group operation distributes over both meets and joins. Similary, a vector lattice (also known as a Riesz space), is a lattice-ordered real vector