International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 12757, 10 pages

On transformation semigroups which are 𝒬-semigroups

S. Nenthein and Y. Kemprasit

Department of Mathematics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

Received 23 December 2005; Revised 6 June 2006; Accepted 22 June 2006

Copyright © 2006 S. Nenthein and Y. Kemprasit. 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.


A semigroup whose bi-ideals and quasi-ideals coincide is called a 𝒬-semigroup. The full transformation semigroup on a set X and the semigroup of all linear transformations of a vector space V over a field F into itself are denoted, respectively, by T(X) and LF(V). It is known that every regular semigroup is a 𝒬-semigroup. Then both T(X) and LF(V) are 𝒬-semigroups. In 1966, Magill introduced and studied the subsemigroup T¯(X,Y) of T(X), where YX and T¯(X,Y)={αT(X,Y)|YαY}. If W is a subspace of V, the subsemigroup L¯F(V,W) of LF(V) will be defined analogously. In this paper, it is shown that T¯(X,Y) is a 𝒬-semigroup if and only if Y=X, |Y|=1, or |X|3, and L¯F(V,W) is a 𝒬-semigroup if and only if (i) W=V, (ii) W={0}, or (iii) F=2, dimFV=2, and dimFW=1 .