Rank of some order-preserving transformation semigroups and combinatorics

Let Cn be the semigroup of all order-preserving and decreasing transformations on Xn = {1, . . . ,n}, under its natural order and let N(Cn) be the set of all nilpotent elements of Cn. Let fix(α) = {x ∈ Xn : xα = x} for any transformation α, and let Cn,Y = {α ∈ Cn : fix(α) =Y} for any subset Y of Xn. In this thesis, we obtain formulas for the number of m-potent elements of N(Cn) and Cn,Y . Furthermore, we determine the structure of all the maximum nilpotent subsemigroups of Cn with their cardinalities and ranks. Keywords: Tranformation semigroups, m-potent, generating set, rank.