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

 Algebra Discrete Math., 2011, Volume 12, Issue 2, Pages 38–52 (Mi adm127)

RESEARCH ARTICLE

On a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point

Ivan Chuchman

Department of Mechanics and Mathematics, Ivan Franko Lviv National University, Universytetska 1, Lviv, 79000, Ukraine

Abstract: In this paper we study the semigroup $\mathfrak{IC}(I,[a])$ ($\mathfrak{IO}(I,[a])$) of closed (open) connected partial homeomorphisms of the unit interval $I$ with a fixed point $a\in I$. We describe left and right ideals of $\mathfrak{IC}(I,[0])$ and the Green's relations on $\mathfrak{IC}(I,[0])$. We show that the semigroup $\mathfrak{IC}(I,[0])$ is bisimple and every non-trivial congruence on $\mathfrak{IC}(I,[0])$ is a group congruence. Also we prove that the semigroup $\mathfrak{IC}(I,[0])$ is isomorphic to the semigroup $\mathfrak{IO}(I,[0])$ and describe the structure of a semigroup $\mathfrak{II}(I,[0])=\mathfrak{IC}(I,[0])\sqcup\mathfrak{IO}(I,[0])$. As a corollary we get structures of semigroups $\mathfrak{IC}(I,[a])$ and $\mathfrak{IO}(I,[a])$ for an interior point $a\in I$.

Keywords: Semigroup of bijective partial transformations, symmetric inverse semigroup, semigroup of homeomorphisms, group congruence, bisimple semigroup.

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Bibliographic databases:
MSC: 20M20,54H15, 20M18
Revised: 22.09.2011
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Citation: Ivan Chuchman, “On a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point”, Algebra Discrete Math., 12:2 (2011), 38–52

Citation in format AMSBIB
\Bibitem{Chu11} \by Ivan Chuchman \paper On a semigroup of closed connected partial homeomorphisms of the unit interval with a~fixed point \jour Algebra Discrete Math. \yr 2011 \vol 12 \issue 2 \pages 38--52 \mathnet{http://mi.mathnet.ru/adm127} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2952900} \zmath{https://zbmath.org/?q=an:1258.20051}