Green’s Relations and Natural Partial Order on the Regular Subsemigroup of Transformations Preserving an Equivalence Relation and Fixed a Cross-Section

Nares Sawatraksa, Chaiwat Namnak, Kritsada Sangkhanan

Abstract


Let $X$ be an arbitrary nonempty set and $T(X)$ the full transformation semigroup on a set $X$. For an equivalence relation $E$ on $X$ and a cross-section $R$ of the partition $X/E$ induced by $E$, let \[ T_{E}(X, R) = \{\alpha \in T(X) : R\alpha = R \mbox{ and } \forall x,y \in X, (x, y) \in E \Rightarrow (x\alpha, y\alpha) \in E \}. \] Then the set $Reg(T_{E}(X, R))$ of all regular elements of $T_{E}(X, R)$ is a regular subsemigroup of $T(X)$. In this paper, we describe Green's relations for elements of the semigroup $Reg(T_{E}(X, R))$. Also, we discuss the natural partial order on semigroup of $Reg(T_{E}(X, R))$ and characterize when two elements of $Reg(T_{E}(X, R))$ are related under this order.

Full Text: PDF

Refbacks

  • There are currently no refbacks.


The Thai Journal of Mathematics organized and supported by The Mathematical Association of Thailand and Thailand Research Council and the Center for Promotion of Mathematical Research of Thailand (CEPMART).

Copyright 2020 by the Mathematical Association of Thailand.

All rights reserve. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission of the Mathematical Association of Thailand.

|ISSN 1686-0209|