On factorisations and generators in transformation semigroups

AYIK G., Ayik H., HOWIE J.

SEMIGROUP FORUM, vol.70, no.2, pp.225-237, 2005 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 70 Issue: 2
  • Publication Date: 2005
  • Doi Number: 10.1007/s00233-004-0145-x
  • Journal Name: SEMIGROUP FORUM
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.225-237
  • Çukurova University Affiliated: Yes

Abstract

It is shown that the classical decomposition of permutations into disjoint cycles can be extended to more general mappings by means of path-cycles, and an algorithm is given to obtain the decomposition. The device is used to obtain information about generating sets for the semigroup of all singular selfmaps of X-n = {1, 2,..., n}. Let T-n,T-r = S-n boolean OR K-n,K-r, where S-n is the symmetric group and K-n,K-r is the set of maps alpha : X-n --> X-n such that \im((alpha)\ <= r. The smallest number of elements of K-n,K-r which, together with S-n, generate T-n,T-r is p(r)(n), the number of partitions of n with r terms.