On Elements of Finite Semigroups of Order-Preserving and Decreasing Transformations

For n ∈ N , let C n be the semigroup of all order-preserving and decreasing transformations on X n = { 1 , … , n } , under its natural order, and let N ( C n ) be the set of all nilpotent elements of C n and let Fix ( α ) = { x ∈ X n : x α = x } for any transformation α . An element a of a finite semigroup is called m -potent ( m -nilpotent) element if a m + 1 = a m ( a m = 0 ) and a , a 2 , … , a m are distinct. In this paper, we obtain a formulae for the number of m -nilpotent elements and so the number of m -potent elements in N ( C n ) for 1 ≤ m ≤ n - 1 . Moreover, for any subset Y of X n , we obtain a formulae for the number of m -potent elements of C n , Y = { α ∈ C n : Fix ( α ) = Y } .