Algebra in the superextensions of semilattices

Details Algebra in the superextensions of semilattices Algebra in the superextensions of semilattices

2010-12-11

arxiv.org/abs/1012.2488v2

Mathematics

Group Theory

10 pages

Scientific paper

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.

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