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International Journal of Mathematics and Mathematical Sciences
Volume 2016, Article ID 2759090, 7 pages
http://dx.doi.org/10.1155/2016/2759090
Research Article

Natural Partial Orders on Transformation Semigroups with Fixed Sets

Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand

Received 20 April 2016; Revised 1 July 2016; Accepted 5 July 2016

Academic Editor: Pentti Haukkanen

Copyright © 2016 Yanisa Chaiya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a nonempty set. For a fixed subset of , let be the set of all self-maps on which fix all elements in . Then is a regular monoid under the composition of maps. In this paper, we characterize the natural partial order on and this result extends the result due to Kowol and Mitsch. Further, we find elements which are compatible and describe minimal and maximal elements.