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 $\phi (X)$ 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.

Détail

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

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