IJPAM: Volume 87, No. 1 (2013)

REGULARITY IN SEMIGROUPS OF TRANSFORMATIONS
WITH INVARIANT SETS

Wanida Choomanee1, Preeyanuch Honyam2, Jintana Sanwong3
1,2,3Department of Mathematics
Faculty of Science
Chiang Mai University
Chiangmai, 50200, THAILAND


Abstract. Let $T(X)$ be the semigroup of all transformations on a set $X$. For a fixed nonempty subset $Y$ of $X$, let

\begin{displaymath}S(X,Y) = \{\alpha\in T(X): Y\alpha\subseteq Y\}.\end{displaymath}

Then $S(X,Y)$ is a semigroup of total transformations on $X$ which leave a subset $Y$ of $X$ invariant. In this paper, we characterize left regular, right regular and intra-regular elements of $S(X,Y)$ and consider the relationships between these elements. Moreover, we count the number of left regular elements of $S(X,Y)$ when $X$ is a finite set.

Received: June 4, 2013

AMS Subject Classification: 20M20

Key Words and Phrases: transformation semigroups, left regular element, right regular element, intra-regular element

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DOI: 10.12732/ijpam.v87i1.9 How to cite this paper?
Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 87
Issue: 1