### Algebra in 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.

#### Keywords

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

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