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 $\phi (X)$ on $X$ as closed subsemigroups. We prove that $\upsilon (X)$ is a semilattice iff $\lambda (X)$ is a semilattice iff $\phi (X)$ 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.