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.

Here we study the principal left k-radicals of a semiring with semilattice additive reduct and characterize semirings which are disjoint union via −→ transitive closure l ∞ relation −→l on S, given by for a, b ∈ ⇔ bn Sa some n N.

Khutoretskii’s Theorem states that the Rogers semilattice of any family of c.e. sets has either at most one or infinitely many elements. A lemma in the inductive step of the proof shows that no Rogers semilattice can be partitioned into a principal ideal and a principal filter. We show that such a partitioning is possible for some family of d.c.e. sets. In fact, we construct a family of c.e. se...