Regularity of semigroups of transformations with restricted range preserving an alternating orientation order

It is well known that the transformation semigroup on a nonempty set $X$, which is denoted by $T(X)$, is regular, but its subsemigroups do not need to be. Consider a finite ordered set $X=(X;\leq)$ whose order forms a path with alternating orientation. For a nonempty subset $Y$ of $X$, two subsemigroups of $T(X)$ are studied. Namely, the semigroup $OT(X,Y)=\{\alpha\in T(X)\mid \alpha~\text{is order-preserving and }X\alpha\subseteq Y\}$ and the semigroup $OS(X,Y)=\{\alpha\in T(X)\mid\alpha$ is order-preserving and $Y\alpha \subseteq Y\}$. In this paper, we characterize ordered sets having a coregular semigroup $OT(X,Y)$ and a coregular semigroup $OS(X,Y)$, respectively. Some characterizations of regular semigroups $OT(X,Y)$ and $OS(X,Y)$ are given. We also describe coregular and regular elements of both $OT(X,Y)$ and $OS(X,Y)$.