Algebra in superextensions of semilattices

• Datum: 2018
• Hauptverfasser: Banakh, Taras, Gavrylkiv, Volodymyr
• Format: Artikel
• Sprache: English
• Veröffentlicht: Lugansk National Taras Shevchenko University 2018
• Schlagworte: semilattice; band; commutative semigroup; the space of upfamilies; the space of filters; the space of maximal linked systems; superextension; 06A12; 20M10
• Online Zugang: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/690
• Назва журналу: Algebra and Discrete Mathematics

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.