This is the first part of a series of two abstract, the second one being by Daniel McNeill.If X is any topological space, its collection of opens sets O(X) is a complete distributive lattice and also a Heyting algebra. When X is equipped with a distinguished basis D(X) for its topology, closed under finite meets and joins, one can investigate situations where D(X) is also a Heyting subalgebra of O(X).Recall that X is a spectral space if it is compact and T0, its collection D(X) of compact open subsets forms a basis which is closed under finite intersections and unions, and X is sober. 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(X) 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(X) is a Heyting subalgebra of O(X).We work with lattice-ordered Abelian groups and vector spaces. Using Mundici’s Gamma-functor the results can be rephrased in terms of MV-algebras, the algebraic semantics of Lukasiewicz infinite-valued propositional logic.Let (G,u) be a finitely presented vector lattice (or Q-vector lattice, or l-group) G equipped with a distinguished strong order unit u. It turns out that Spec(G,u), i.e. the the space of prime ideals of (G,u) topologised with the hull-kernel topology, is a compact spectral space. Our first main result states that the collection D(Spec(G,u)) of compact open subsets of Spec(G,u) is a Heyting subalgebra of the Heyting algebra of open subsets O(Spec(G,u)).As a consequence, we also prove that the subspace Min(G,u) of minimal prime ideals of G is a Boolean space, i.e. a compact Hausdorff space whose clopen sets form a basis for the topology.Further, for any fixed maximal ideal m of G, the set l(m) of prime ideals of G contained in m, equipped with the subspace topology, is a spectral space, and the subspace Min(l(m)) of l(m) is a Boolean space.
Some remarks on divisible polyhedral MV-algebras
