Weakly regular semigroups of isotone transformations

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

Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 4, pp. 145-165

Résumé

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 $y$ for $y\in Y$, $z\in Z$; (4) $X=Y\cup Z$, where $y_0\in Y$, $z_0\in Z$, and $y_0$ for $z\in Z$, $y$ for $y\in Y$; (5) $X=\{a,c\}\cup B$, where $a$ for $b\in B$; (6) $X=\{1,2,3,4,5,6\}$, where $14$, $15$, $25$, $26$, $34$, $36$. 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 $x,y\in X$.