Summary: 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 and 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.