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).

Key Words:

• Idempotent rank
• Order preserving transformation
• Orientation preserving transformation

2000 Mathematics Subject Classification:

• 20M20

ACKNOWLEDGMENTS

The author would like to thank Lauren Ferguson and Mei Yang for their proofreading and comments. Also, the author is deeply grateful to the referee(s) for his/her valuable comments and suggestions.

This work is supported by Natural Science Fund of Guizhou (No. [2010] 3174).

Notes

Communicated by V. Gould.