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 $\phi(X)$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2010
- DOI:
- 10.48550/arXiv.1012.2488
- arXiv:
- arXiv:1012.2488
- Bibcode:
- 2010arXiv1012.2488B
- Keywords:
-
- Mathematics - Group Theory;
- Mathematics - Rings and Algebras;
- 06A12;
- 20M10
- E-Print:
- 10 pages