Combinatorial results for semigroups of orientation-preserving transformations

Let Xn denote the chain 1, 2,. . . , n under its natural order. We denote the semigroups consisting of all order-preserving transformations and all orientation-preserving transformations on Xn by On and OPn, respectively. We denote by E (U) the set of all idempotents of a subset U of a semigroup S. In this paper, we first determine the cardinalities of Er (On) = α ∈ E (On): |im (α) | = |fix (α) | = r, E ∗ r (On) = α ∈ Er (On): 1, n ∈ fix (α), Er (OPn) = α ∈ E (OPn): |fix (α) | = r, E ∗ r (OPn) = α ∈ Er (OPn): n ∈ fix (α) (1 ≤ r ≤ n) and then, by using these results, we determine the numbers of idempotents in On and OPn by a new method. Let OP− n denote the semigroup of all orientation-preserving and order-decreasing transformations on Xn. Moreover, we determine the cardinalities of OP− n, OP− n, Y = α ∈ OP− n: fix (α) = Y for any nonempty subset Y of Xn and OP− n, r = α ∈ OP− n: |fix (α) | = r for 1 ≤ r ≤ n. Also, we determine the number of idempotents in OP− n and the number of nilpotents in OP− n.