Regularity conditions for semigroups of isotone transformations of countable chains

Let $\Gamma$ be a linearly ordered set (a chain), $O(\Gamma)$ be the semigroup of all isotone transformations of $\Gamma$ (i.e., order-preserving transformations). We find some necessary and some sufficient conditions on the chain $\Gamma$ for the semigroup $O(\Gamma)$ to be regular. For example, if $\Gamma$ is a complete chain with the maximal element and the minimal one, then $O(\Gamma)$ is regular. In particular, $O(\Gamma)$ is regular if $\Gamma$ is finite. We find necessary and sufficient conditions for the regularity of $O(\Gamma)$ in the case where $\Gamma$ is countable.