Algebra in the 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.

arXiv comment10 pages

References in corpus (2)

• Right-topological semigroup operations on inclusion hyperspaces
• Algebra in superextension of groups, II: cancelativity and centers

Cited by in corpus (2)

• Algebra in superextensions of inverse semigroups
• Automorphism groups of superextensions of finite monogenic semigroups