Algebra in superextensions of semilattices

Abstract

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