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 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 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...

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.

If P is an upper semilattice whose Hasse diagram is a tree and whose cartesian powers are Macaulay, it is shown that Hasse diagram of P is actually a spider in which all the legs have the same length.

We prove that the semirings of 1-preserving and of 0,1preserving endomorphisms of a semilattice are always subdirectly irreducible and we investigate under which conditions they are simple. Subsemirings are also investigated in a similar way.

Let S be a multiplicatively idempotent congruence-simple semiring. We show that $$|S|=2$$ if has absorbing element. also prove is finite then either or $$S\cong {{\,\textrm{End}\,}}(L)$$ $$S^{op}\cong where L the 2-element semilattice. It seems to an open question, whether can infinite at all.