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


YAĞCI M. , Korkmaz E.

BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, vol.43, no.5, pp.3863-3870, 2020 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 43 Issue: 5
  • Publication Date: 2020
  • Doi Number: 10.1007/s40840-020-00899-7
  • Title of Journal : BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY
  • Page Numbers: pp.3863-3870

Abstract

For n is an element of 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 (alpha) = {x is an element of X-n : x alpha = x} for any transformation alpha. 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,C- Y = {alpha is an element of C-n : Fix (alpha) = Y}.