FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(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

View as HTML     View as gif image

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., $x$£ y Þ xa £ ya for all $x,y$Î 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) $X\; =\; Y$È Z, where 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, ₀ for $y$Î Y; (5) $X\; =\; \{a,c\}$È B, where for $b$Î B; (6) $X\; =\; \{1,2,3,4,5,6\}$, where , , , , , . 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 $x$£ y for all $x,y$Î X.

Location: http://mech.math.msu.su/~fpm/eng/k1112/k114/k11409h.htm
Last modified: July 2, 2012