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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2012, Volume 13, Issue 1, Pages 26–42 (Mi adm63)

RESEARCH ARTICLE

Algebra in superextensions of semilattices

Taras Banakhab, Volodymyr Gavrylkivc

a Ivan Franko National University of Lviv, Ukraine
b Jan Kochanowski University, Kielce, Poland
c Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine

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–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.

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

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Bibliographic databases:
MSC: 06A12, 20M10
Revised: 19.01.2012
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Citation: Taras Banakh, Volodymyr Gavrylkiv, “Algebra in superextensions of semilattices”, Algebra Discrete Math., 13:1 (2012), 26–42

Citation in format AMSBIB
\Bibitem{BanGav12} \by Taras~Banakh, Volodymyr~Gavrylkiv \paper Algebra in superextensions of semilattices \jour Algebra Discrete Math. \yr 2012 \vol 13 \issue 1 \pages 26--42 \mathnet{http://mi.mathnet.ru/adm63} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2963823} \zmath{https://zbmath.org/?q=an:06120560} 

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This publication is cited in the following articles:
1. Taras Banakh, Volodymyr Gavrylkiv, “Algebra in superextensions of inverse semigroups”, Algebra Discrete Math., 13:2 (2012), 147–168
2. Taras Banakh, Volodymyr Gavrylkiv, “Characterizing semigroups with commutative superextensions”, Algebra Discrete Math., 17:2 (2014), 161–192
3. T. O. Banakh, V. M. Gavrylkiv, “On structure of the semigroups of k-linked upfamilies on groups”, Asian-Eur. J. Math., 10:4 (2017), 1750083
4. V. Gavrylkiv, “Semigroups of centered upfamilies on groups”, Lobachevskii J. Math., 38:3, SI (2017), 420–428
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