Algebra in superextensions of semilattices

Taras Banakh ; Volodymyr Gavrylkiv

Algebra and discrete mathematics, Tome 13 (2012) no. 1, pp. 26-42

Résumé

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–Čech 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.

Détail

Keywords: semilattice, band, commutative semigroup, the space of upfamilies, the space of filters, the space of maximal linked systems, superextension.