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Fundam. Prikl. Mat., 2012, Volume 17, Issue 4, Pages 145–165 (Mi fpm1426)  

This article is cited in 1 scientific paper (total in 1 paper)

Weakly regular semigroups of isotone transformations

V. I. Kim, I. B. Kozhukhov, V. A. Yaroshevich

Moscow State Institute of Electronic Technology (Technical University)

Abstract: Let $X$ be a partially ordered set and $O(X)$ be the semigroup of all mappings $X\to X$ that preserve the order, i.e., $x\leq y\Longrightarrow x\alpha\leq y\alpha$ for all $x,y\in X$. It is proved that the semigroup $O(X)$ is weakly regular in the wide sense if and only if at least one of the following conditions holds: (1) $X$ is a quasi-complete chain; (2) the elements of $X$ are not comparable pairwise; (3) $X=Y\cup Z$, where $y<z$ for $y\in Y$, $z\in Z$; (4) $X=Y\cup Z$, where $y_0\in Y$, $z_0\in Z$, and $y_0<z$ for $z\in Z$, $y<z_0$ for $y\in Y$; (5) $X=\{a,c\}\cup B$, where $a<b<c$ for $b\in B$; (6) $X=\{1,2,3,4,5,6\}$, where $1<4$, $1<5$, $2<5$, $2<6$, $3<4$, $3<6$. Moreover, if $X$ is a quasi-ordered set but not partially ordered, then the semigroup $O(X)$ is weakly regular in the wide sense if and only if $x\leq y$ for all $x,y\in X$.

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English version:
Journal of Mathematical Sciences (New York), 2013, 191:5, 694–708

Document Type: Article
UDC: 512.534.3

Citation: V. I. Kim, I. B. Kozhukhov, V. A. Yaroshevich, “Weakly regular semigroups of isotone transformations”, Fundam. Prikl. Mat., 17:4 (2012), 145–165; J. Math. Sci., 191:5 (2013), 694–708

Citation in format AMSBIB
\Bibitem{KimKozYar12}
\by V.~I.~Kim, I.~B.~Kozhukhov, V.~A.~Yaroshevich
\paper Weakly regular semigroups of isotone transformations
\jour Fundam. Prikl. Mat.
\yr 2012
\vol 17
\issue 4
\pages 145--165
\mathnet{http://mi.mathnet.ru/fpm1426}
\transl
\jour J. Math. Sci.
\yr 2013
\vol 191
\issue 5
\pages 694--708
\crossref{https://doi.org/10.1007/s10958-013-1353-2}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84884990378}


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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. Yaroshevich V.A., “O regulyarnykh polugruppakh izotonnykh preobrazovanii chastichno uporyadochennykh mnozhestv”, Vestnik moskovskoi gosudarstvennoi akademii delovogo administrirovaniya. seriya: ekonomika, 2012, no. 4, 130–134  mathscinet  elib
  • Фундаментальная и прикладная математика
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