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

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Let X be a partially ordered set and O(X) be the semigroup of all mappings X → 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 y < z for y Î Y, z Î Z; (4) X = Y È Z, where y0 Î Y, z0 Î Z, and y0 < z for z Î Z, y < z0 for y Î Y; (5) X = {a,c} È B, where a < b < c for b Î 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 £ y for all x,y Î X.

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