Regular Transformation Semigroups on Some Dictionary Chains

Denote by OT(X) the full order-preserving transformation semigroup on a poset X. The following results are known. If X is any nonempty subset of $\mathbb{Z}$ with the natural order, then OT ( X ) is a regular semigroup, that is, for every $\alpha\in OT(X), \alpha = \alpha\beta\alpha$ for some $\beta\in OT(X)$. If $\leq_d$ is the dictionary partial order on $X \times X$ where X is a nonempty subset of $\mathbb{Z}$, then $OT(X\times X, \leq_d)$ is regular if nd only if X is finite. By using these two known results, we extend the second one to the semigroup $OT(X\times Y, \leq_d)$ where X and Y are nonempty subsets of $\mathbb{Z}$. It is shown that $OT(X\times Y, \leq_d)$ is regular if and only if $|X| = 1$ or Y is finite.