On the Ranks of Certain Semigroups of Orientation Preserving Transformations

Abstract

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] 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).