International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 489674, 14 pages
doi:10.1155/2011/489674
Research Article

On Maximal Subsemigroups of Partial Baer-Levi Semigroups

1Department of Mathematics, Chiang Mai University, Chiangmai 50200, Thailand
2Material Science Research Center, Faculty of Science, Chiang Mai University, Chiangmai 50200, Thailand

Received 20 September 2010; Revised 14 January 2011; Accepted 28 February 2011

Academic Editor: Robert Redfield

Copyright © 2011 Boorapa Singha and Jintana Sanwong. 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

Suppose that ๐‘‹ is an infinite set with | ๐‘‹ | โ‰ฅ ๐‘ž โ‰ฅ โ„ต 0 and ๐ผ ( ๐‘‹ ) is the symmetric inverse semigroup defined on ๐‘‹ . In 1984, Levi and Wood determined a class of maximal subsemigroups ๐‘€ ๐ด (using certain subsets ๐ด of ๐‘‹ ) of the Baer-Levi semigroup ๐ต ๐ฟ ( ๐‘ž ) = { ๐›ผ โˆˆ ๐ผ ( ๐‘‹ ) โˆถ dom ๐›ผ = ๐‘‹ and | ๐‘‹ โงต ๐‘‹ ๐›ผ | = ๐‘ž } . Later, in 1995, Hotzel showed that there are many other classes of maximal subsemigroups of ๐ต ๐ฟ ( ๐‘ž ) , but these are far more complicated to describe. It is known that ๐ต ๐ฟ ( ๐‘ž ) is a subsemigroup of the partial Baer-Levi semigroup ๐‘ƒ ๐‘† ( ๐‘ž ) = { ๐›ผ โˆˆ ๐ผ ( ๐‘‹ ) โˆถ | ๐‘‹ โงต ๐‘‹ ๐›ผ | = ๐‘ž } . In this paper, we characterize all maximal subsemigroups of ๐‘ƒ ๐‘† ( ๐‘ž ) when | ๐‘‹ | > ๐‘ž , and we extend ๐‘€ ๐ด to obtain maximal subsemigroups of ๐‘ƒ ๐‘† ( ๐‘ž ) when | ๐‘‹ | = ๐‘ž .