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Algebra Discrete Math., 2016, Volume 21, Issue 2, Pages 163–183 (Mi adm561)  

RESEARCH ARTICLE

On a semitopological polycyclic monoid

Serhii Bardyla, Oleg Gutik

Faculty of Mathematics, National University of Lviv, Universytetska 1, Lviv, 79000, Ukraine

Abstract: We study algebraic structure of the $\lambda$-polycyclic monoid $P_{\lambda}$ and its topologizations. We show that the $\lambda$-polycyclic monoid for an infinite cardinal $\lambda\geqslant 2$ has similar algebraic properties so has the polycyclic monoid $P_n$ with finitely many $n\geqslant 2$ generators. In particular we prove that for every infinite cardinal $\lambda$ the polycyclic monoid $P_{\lambda}$ is a congruence-free combinatorial $0$-bisimple $0$-$E$-unitary inverse semigroup. Also we show that every non-zero element $x$ is an isolated point in $(P_{\lambda},\tau)$ for every Hausdorff topology $\tau$ on $P_{\lambda}$, such that $(P_{\lambda},\tau)$ is a semitopological semigroup, and every locally compact Hausdorff semigroup topology on $P_\lambda$ is discrete. The last statement extends results of the paper [33] obtaining for topological inverse graph semigroups. We describe all feebly compact topologies $\tau$ on $P_{\lambda}$ such that $(P_{\lambda},\tau)$ is a semitopological semigroup and its Bohr compactification as a topological semigroup. We prove that for every cardinal $\lambda\geqslant 2$ any continuous homomorphism from a topological semigroup $P_\lambda$ into an arbitrary countably compact topological semigroup is annihilating and there exists no a Hausdorff feebly compact topological semigroup which contains $P_{\lambda}$ as a dense subsemigroup.

Keywords: inverse semigroup, bicyclic monoid, polycyclic monoid, free monoid, semigroup of matrix units, topological semigroup, semitopological semigroup, Bohr compactification, embedding, locally compact, countably compact, feebly compact.

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Bibliographic databases:
MSC: Primary 22A15, 20M18; Secondary 20M05, 22A26, 54A10, 54D30, 54D35, 54D45, 54H11
Received: 29.01.2016
Revised: 16.02.2016
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Citation: Serhii Bardyla, Oleg Gutik, “On a semitopological polycyclic monoid”, Algebra Discrete Math., 21:2 (2016), 163–183

Citation in format AMSBIB
\Bibitem{BarGut16}
\by Serhii~Bardyla, Oleg~Gutik
\paper On a semitopological polycyclic monoid
\jour Algebra Discrete Math.
\yr 2016
\vol 21
\issue 2
\pages 163--183
\mathnet{http://mi.mathnet.ru/adm561}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3537444}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000382847700002}


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