Algebra in superextensions of semilattices

Abstract

Given a semilattice X we study the algebraic properties of the semigroup υ(X) of upfamilies on X. The semigroup υ(X) contains the Stone-ˇCech extension β(X), the superextension λ(X), and the space of filters φ(X) on X as closed subsemigroups. We prove that υ(X) is a semilattice iff λ(X) is a semilattice iff φ(X) is a semilattice iff the semilattice X is finite and linearly ordered. We prove that the semigroup β(X) is a band if and only if X has no infinite antichains, and the semigroup λ(X) is commutative if and only if X is a bush with finite branches.