For k randomly chosen subsets of [n] = {1,2, ,n} we consider the probability that the Boolean algebra, distributive lattice, and meet semilattice which they generate are respectively free, or all of 2*. In each case we describe a threshold function for the occurrence of these events. The threshold functions for freeness are close to their theoretical maximum values. 1 §

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 ...

We prove that the variety of nuclear implicative semilattices is locally finite, thus generalizing Diego’s Theorem. The key ingredients our proof include coloring technique and construction universal models from modal logic. For this we develop duality theory for finite semilattices, Köhler duality. main result remains true bounded give an alternative Theorem, provide explicit description free ...

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 show that for every quasivariety K of relational structures there is a semilattice S with operators such that the lattice of quasiequational theories of K (the dual of the lattice of sub-quasivarieties of K) is isomorphic to Con(S,+, 0, F). As a consequence, new restrictions on the natural quasi-interior operator on lattices of quasi-equational theories are found. 1. Motivation and terminolo...

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.

Recent results of Bucciarelli show that the semilattice of degrees of parallelism of firstorder boolean functions in PCF has both infinite chains and infinite antichains. By considering a simple subclass of Sieber’s sequentiality relations, we identify levels in the semilattice and derive inexpressibility results concerning functions on different levels. This allows us to further explore the st...

We introduce and study some natural operations on the structure of finite labeled forests which is of central interest for extending the difference hierarchy to the case of partitions. It is shown that the corresponding quotient-algebra modulo the so called h-equivalence is the simplest nontrivial semilattice with discrete closures. The algebra is also characterized as a free algebra in some qu...