Download PDFOpen PDF in browserSpectra of finitely presented latticeordered Abelian groups and MValgebras, part 15 pages•Published: July 28, 2014AbstractThis 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 hullkernel 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 latticeordered Abelian groups and vector spaces. Using Mundici’s Gammafunctor the results can be rephrased in terms of MValgebras, the algebraic semantics of Lukasiewicz infinitevalued propositional logic. Let (G,u) be a finitely presented vector lattice (or Qvector lattice, or lgroup) 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 hullkernel 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. Keyphrases: lattice ordered abelian group, lukasiewicz logic, mv algebra, spectral space, stone duality, strong order unit, vector lattice In: Nikolaos Galatos, Alexander Kurz and Constantine Tsinakis (editors). TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic, vol 25, pages 148152.
