(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 4, PAGES 145-165

## Weakly regular semigroups of isotone transformations

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

Abstract

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Let $X$ be a partially ordered set and $O\left(X\right)$ be the semigroup of all mappings $X \to X$ that preserve the order, i.e., $x$£ y Þ xa £ ya for all $x,y$Î X. It is proved that the semigroup $O\left(X\right)$ 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) $X = Y$È Z, where $y < z$ for $y$Î Y, $z$Î Z; (4) $X = Y$È Z, where $y$0 Î Y, $z$0 Î Z, and $y$0 < z for $z$Î Z, $y < z$0 for $y$Î Y; (5) $X = \left\{a,c\right\}$È B, where $a < b < c$ for $b$Î B; (6) $X = \left\{1,2,3,4,5,6\right\}$, 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 $O\left(X\right)$ is weakly regular in the wide sense if and only if $x$£ y for all $x,y$Î X.

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