We prove that for every distributive ∨, 0-semilattice S, there are a meet-semilattice P with zero and a map µ : P × P → S such that µ(x, z) ≤ µ(x, y) ∨ µ(y, z) and x ≤ y implies that µ(x, y) = 0, for all x, y, z ∈ P , together with the following conditions: (P1) µ(v, u) = 0 implies that u = v, for all u ≤ v in P. (P2) For all u ≤ v in P and all a, b ∈ S, if µ(v, u) ≤ a ∨ b, then there are a pos...

We say that a 〈∨, 0〉-semilattice S is conditionally co-Brouwerian, if (1) for all nonempty subsets X and Y of S such that X ≤ Y (i.e., x ≤ y for all 〈x, y〉 ∈ X × Y ), there exists z ∈ S such that X ≤ z ≤ Y , and (2) for every subset Z of S and all a, b ∈ S, if a ≤ b ∨ z for all z ∈ Z, then there exists c ∈ S such that a ≤ b ∨ c and c ≤ Z. By restricting this definition to subsets X , Y , and Z ...

In general, the tensor product, A ⊗ B, of the lattices A and B with zero is not a lattice (it is only a join-semilattice with zero). If A ⊗ B is a capped tensor product, then A ⊗ B is a lattice (the converse is not known). In this paper, we investigate lattices A with zero enjoying the property that A ⊗ B is a capped tensor product, for every lattice B with zero; we shall call such lattices ame...

There are various ways to obtain distributive semilattices from other mathematical objects. Two of them are the following; we refer to Section 1 for more precise definitions. A dimension group is a directed, unperforated partially ordered abelian group with the interpolation property, see also K.R. Goodearl [6]. With a dimension group G we can associate its semilattice of compact ( = finitely g...

We prove that for every distributive 〈∨, 0〉-semilattice S, there are a meet-semilattice P with zero and a map μ : P × P → S such that μ(x, z) ≤ μ(x, y)∨μ(y, z) and x ≤ y implies that μ(x, y) = 0, for all x, y, z ∈ P , together with the following conditions: (P1) μ(v, u) = 0 implies that u = v, for all u ≤ v in P . (P2) For all u ≤ v in P and all a,b ∈ S, if μ(v, u) ≤ a ∨ b, then there are a pos...

We study chains in an H-closed topological partially ordered space. We give sufficient conditions for a maximal chain L in an H-closed topological partially ordered space such that L contains a maximal (minimal) element. Also we give sufficient conditions for a linearly ordered topological partially ordered space to be H-closed. We prove that any H-closed topological semilattice contains a zero...

This is a version from 29 Sept 2003 of the paper published under the same name in Theoretical Computer Science 316 (2004) 297–321. The double powerlocale P(X) (found by composing, in either order, the upper and lower powerlocale constructions PU and PL) is shown to be isomorphic in [Loc,Set] to the double exponential SS X where S is the Sierpiński locale. Further PU (X) and PL(X) are shown to b...

This article studies commutative orders, that is, commutative semigroups having a semigroup of quotients. In a commutative order S, the square-cancellable elements S(S) constitute a well-behaved separable subsemigroup. Indeed, S(S) is also an order and has a maximum semigroup of quotients R, which is Clifford. We present a new characterisation of commutative orders in terms of semilattice decom...

Considering the lattice of properties of a physical system, it has been argued elsewhere that – to build a calculus of propositions having a well-behaved notion of disjunction (and implication) – one should consider a very particular frame completion of this lattice. We show that the pertinent frame completion is obtained as sheafification of the presheaves on the given meet-semilattice with re...