We investigate the class of reflexive graphs that admit semilattice polymorphisms, and in doing so, give an algebraic characterisation of chordal graphs. In particular, we show that a graph G is chordal if and only if it has a semilattice polymorphism such that G is a subgraph of the comparability graph of the semilattice. Further, we find a new characterisation of the leafage of a chordal grap...

A modular semilattice is a semilattice S in which w > a A ft implies that there exist i,jeS such that x > a. y > b and x A y = x A w. This is equivalent to modularity in a lattice and in the semilattice of ideals of the semilattice, and the condition implies the Kurosh-Ore replacement property for irreducible elements in a semilattice. The main results provide extensions of the classical charac...

While every finite lattice-based algebra is dualisable, the same is not true of semilattice-based algebras. We show that a finite semilattice-based algebra is dualisable if all its operations are compatible with the semilattice operation. We also give examples of infinite semilattice-based algebras that are dualisable. In contrast, we present a general condition that guarantees the inherent non...

Let A := (A,→, 1) be a Hilbert algebra. The monoid of all unary operations on A generated by operations αp : x → (p → x), which is actually an upper semilattice w.r.t. the pointwise ordering, is called the adjoint semilattice of A. This semilattice is isomorphic to the semilattice of finitely generated filters of A, it is subtractive (i.e., dually implicative), and its ideal lattice is isomorph...

A 〈∨, 0〉-semilattice is ultraboolean, if it is a directed union of finite Boolean 〈∨, 0〉-semilattices. We prove that every distributive 〈∨, 0〉-semilattice is a retract of some ultraboolean 〈∨, 0〉-semilattice. This is established by proving that every finite distributive 〈∨, 0〉-semilattice is a retract of some finite Boolean 〈∨, 0〉-semilattice, and this in a functorial way. This result is, in tu...

We find a distributive (∨, 0, 1)-semilattice Sω1 of size א1 that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an order-unit of finite stable rank. We thus obtain solutions to various open problems in ring theory and in lattice theory. In particular: — There is no exchange ring (thus, no von Neumann regular ring and no C*-algebra of real rank zero) with ...

Let S, T be semilattices. Let us assume that if S is upper-bounded, then T is upper-bounded. A map from S into T is said to be a semilattice morphism from S into T if: (Def. 1) For every finite subset X of S holds it preserves inf of X . Let S, T be semilattices. Observe that every map from S into T which is meet-preserving is also monotone. Let S be a semilattice and let T be an upper-bounded ...

As Jonathan Leech has pointed out, many natural examples of inverse semigroups are semilattice ordered under the natural partial order. But there are many interesting examples of semilattice ordered inverse semigroups in which the imposed partial order is not the natural one. In this talk we shall explore the structure and properties of these examples and some other questions related to semilat...

We prove that the endomorphism semiring of a nontrivial semilattice is always subdirectly irreducible and describe its monolith. The endomorphism semiring is congruence simple if and only if the semilattice has both a least and a largest element.

We consider the lower semilattice D of diierences of c.e. sets under inclusion. It is shown that D is not distributive as a semilattice, and that the c.e. sets form a deenable subclass.