Regularity conditions for semigroups of isotone transformations of countable chains

Let Γ be a linearly ordered set (a chain), O(Γ) be the semigroup of all isotone transformations of Γ (i.e., order-preserving transformations). We find some necessary and some sufficient conditions on the chain Γ for the semigroup O(Γ) to be regular. For example, if Γ is a complete chain with a maximal element and a minimal one, then O(Γ) is regular. In particular, O(Γ) is regular if Γ is finite. We find necessary and sufficient conditions for the regularity of O(Γ) in the case where Γ is countable. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 8, pp. 97–104, 2006.