Prime spectra of lattice-ordered abelian groups

Abstract

We prove that for each P-group G, the topological space Spec(G) satisfies a condition ldtu. Generalising a previous construction of Delzell and Madden we show that for each nondenumerable cardinal there is a completely normal spectral space that is not homeomorphic to Spec( G) for any d-group G. We show also that a stronger form of property ldw, called Id, suffices to ensure that a completely normal spectral space is homeomorphic to Spec(G) for some /-group G.

Gödel spaces and perfect MV-algebras

“…Not every completely normal spectral space is a spectral space of some -group. Notice, also, that there exists an -group G (with strong unit) such that the distributive lattice, corresponding to the spectral space Spec(G), is not dual Heyting (or op-Heyting) algebra [7]. Taking into account that the category of MV -algebras is equivalent to the category of -groups with strong unit we conclude that not every MV -space is a Gödel space.…”

“…Notice that the spectral spaces of -groups (also with strong unit was investigated in [7]), are root systems (or in other terminology, completely normal spectral spaces). Not every completely normal spectral space is a spectral space of some -group.…”

The Absolute of a Topological Space and its Application to Abelian l-groups

“…With S(a) := S({a}), this implies that {S(a) ∩ S(a ) | a / ∈ Q, a / ∈ Q } = ∅. Since Spec H is compact in the patch topology [10], it follows that Q and Q have disjoint neighbourhoods. So the condition of Corollary 2 just means that Spec e is separated.…”

The essential cover and the absolute cover of a schematic space

“…Spectral spaces arise as spectra of commutative rings or abelian l-groups [17,10]. By Stone's duality theorem [26], the spectrum of a bounded distributive lattice is also a spectral space.…”

“…Not every spectral space is homeomorphic to the spectrum of an abelian l-group. Some necessary conditions are known [10], but it is still open which spectral spaces actually occur.…”