Weakly Regular Semigroups of Isotone Transformations |
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Authors: | V I Kim I B Kozhukhov V A Yaroshevich |
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Institution: | 1. Moscow Institute of Electronic Engineering, Moscow, Russia
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Abstract: | 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 ? xα ≤ yα for all x, y ∈ 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 ∪ Z, where y < z for y ∈ Y, z ∈ Z; (4) X = Y ∪ Z, where y 0 ∈ Y, z 0 ∈ Z, and y 0 < z for z ∈ Z, y < z0 for y ∈ Y; (5) X = {a, c} ∪ B, where a < b < c for b ∈ 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 ∈ X. |
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