Combinatorial results of collapse for order-preserving and order-decreasing transformations

Abstract

The full transformation semigroup Tn is defined to consist of all functions from Xn = {1, . . . , n} to itself, under the operation of com- position. In [9], for any α in Tn, Howie defined and denoted collapse by c(α) = ⋃ t∈im(α) {tα−1 : |tα−1| ≥ 2}. Let On be the semigroup of all order- preserving transformations and Cn be the semigroup of all order-preserving and decreasing transformations on Xn under its natural order, respectively. Let E(On) be the set of all idempotent elements of On, E(Cn) and N (Cn) be the sets of all idempotent and nilpotent elements of Cn, respectively. Let U be one of {Cn, N (Cn), E(Cn), On, E(On)}. For α ∈ U , we consider the set imc(α) = {t ∈ im(α) : |tα−1| ≥ 2}. For positive integers 2 ≤ k ≤ r ≤ n, we define U (k) = {α ∈ U : t ∈ imc(α) and |tα−1| = k}, U (k, r) = {α ∈ U (k) : ∣ ∣ ⋃ t∈imc (α) tα−1| = r}. The main objective of this paper is to determine |U (k, r)|, and so |U (k)| for some values r and k.