Rank Properties of the Semigroup of Singular Transformations on a Finite Set
Gonca Ayık, Hayrullah Ayık, Yusuf Ünlü, John M Howie · Communications in Algebra · 2008
It is known that the semigroup Sing n of all singular self-maps of X n = {1,2,…, n} has rank n(n − 1)/2. The idempotent rank, defined as the smallest number of idempotents generating Sing n , has the same value as the rank. (See Gomes and Howie, 1987 Gomes , G. M. S. , Howie , J. M. ( 1987 ). On the rank of certain finite semigroups of transformations . Math. Proc. Cambridge Phil. Soc. 101 : 395 – 303 .[Crossref], [Web of Science ®] , [Google Scholar].) Idempotents generating Sing n can be seen as special cases (with m = r = 2) of (m, r)-path-cycles, as defined in Ay\i k et al. (2005 Ayık , G. , Ayık , H. , Howie , J. M. ( 2005 ). On factorisations and generators in transformation semigroups . Semigroup Forum 70 : 225 – 237 .[Crossref], [Web of Science ®] , [Google Scholar]). The object of this article is to show that, for fixed m and r, the (m, r)-rank of Sing n , defined as the smallest number of (m, r)-path-cycles generating Sing n , is once again n(n − 1)/2.