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

This article is cited in 4 scientific papers (total in 4 papers)

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
Received: 05.10.2011
Revised: 19.01.2012
Language:

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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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  mathscinet  zmath
    2. Taras Banakh, Volodymyr Gavrylkiv, “Characterizing semigroups with commutative superextensions”, Algebra Discrete Math., 17:2 (2014), 161–192  mathnet  mathscinet
    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  crossref  mathscinet  zmath  isi  scopus
    4. V. Gavrylkiv, “Semigroups of centered upfamilies on groups”, Lobachevskii J. Math., 38:3, SI (2017), 420–428  crossref  mathscinet  zmath  isi  scopus
  • Algebra and Discrete Mathematics
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