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...

This note makes two observations about lattices of subsemilattices. First, we establish relationship between direct decompositions of such lattices and ordinal sum decompositions of semilattices. Then we give a characterization of the subsemilattice-lattices. Let us recall some terminology. L will always stand for a semilattice, whose operation will be denoted by. a of an arbitrary lattice L is...

Let a finite semilattice S be a chain under its natural order. We show that if a semigroup T divides a semigroup of full order preserving transformations of a finite chain, then so does any semidirect product S o T .

In this paper, we prove that the Birget-Rhodes expansion G̃R of a group G is not a semidirect product of a semilattice by a group but it can be nicely embedded into such a semidirect product.