Ranks and presentations for order-preserving transformations with one fixed point

In the present paper, we consider the semigroup O-n,O-p of all order-preserving full transformations alpha on an n-elements chain X-n, where p is an element of X-n is the only fixed point of alpha. The nilpotent semigroup O-n,O-p was first studied by Ayik et al. in 2011. Moreover, O-n,O-1 is the maximal nilpotent subsemigroup of the Catalan Monoid C-n. Its rank is the difference of the (n - 1)th and the (n - 2) th Catalan number. The aim of the present paper is to provide further fundamental information about the nilpotent semigroup O-n,O-p. We will calculate the rank of O-n,O-p for p > 1 and provide a semigroup presentation for O-n,O-1.