On the Ranks of Certain Semigroups of Orientation Preserving Transformations
Ping Zhao · Communications in Algebra · 2011
In this article, we investigate the ranks of SOP n , SPOP n , and SSPOP n (the semigroups of orientation preserving singular selfmaps, partial, and strictly partial transformations on [n] = {1, 2,…, n}, respectively). Firstly, we show that the rank and idempotent rank of SOP n are n. Secondly, we characterize the structure of the idempotent-generating sets of SPOP n , and prove that the rank and idempotent rank of SPOP n are 2n. Finally, we find that the rank of SSPOP n is n + 1. This research extends the results of Gomes and Howie [4 Gomes , G. M. S. , Howie , J. M. ( 1992 ). On the ranks of certain semigroups of order-preserving transformations . Semigroup Forum 45 : 272 – 282 .[Crossref], [Web of Science ®] , [Google Scholar]] on the ranks of the semigroups O n , PO n , and SPO n (the semigroups of order preserving singular selfmaps, partial, and strictly partial transformations on [n] = {1, 2,…, n}, respectively).