BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, vol.42, no.3, pp.921-932, 2019 (SCI-Expanded)
For nN, let On be the semigroup of all order-preserving transformations on the finite chain Xn={1,...,n}, under its natural order. For any non-empty subset A of Xn, let On(A) and On+(A) be the subsemigroups of all order-preserving and A-decreasing, and of all order-preserving and A-increasing transformations on Xn, respectively. In this paper we obtain formulae for the number of elements and for the number of idempotents in On(A). Moreover, we show that On(A) contains a zero element if and only if 1A, and then we obtain the number of nilpotents in