Regular subsemigroups of the semigroups of transformations preserving a fence

It is well-known that the transformation semigroup T(X) is regular, but their subsemigroups need not be. A fence is an ordered set that the order forms a path with alternating orientation. Consider X as the base set of a fence X = (X;=). Two subsemigroups of T(X) are studied. Namely, the semigroup DT(X) of all order-decreasing self-mappings of X and the semigroup OT(X) of all order-preserving self-mappings of X. In this paper, we obtain that DT(X) is a coregular subsemigroup of T(X). A characterization of regular subsemigroups OT(X) of T(X) is given, that is, OT(X) is regular if and only if |X| <= 4. Finally, we discuss the regularity of elements in OT(X).