E-Inversive Elements in Some Semigroups of Transformations that Preserve Equivalence

Nares Sawatraksa, Punyapat Kammoo, Chaiwat Namnak


Let $X$ be a 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 \[ T_{E^{*}}(X) = \{\alpha \in T(X) : \forall x, y \in X, (x, y) \in E \Leftrightarrow (x\alpha, y\alpha) \in E \} \text{ and } \] \[ T_{E}(X, R) = \{\alpha \in T(X) : R\alpha = R \mbox{ and } \forall x,y \in X, (x, y) \in E \Rightarrow (x\alpha, y\alpha) \in E \}. \] Then $T_{E^{*}}(X)$ and $T_{E}(X, R)$ are subsemigroups of $T(X)$. In this paper, we describe the $E$-inversive elements of $T_{E^{*}}(X)$ and $T_{E}(X, R)$. We also show that $T_{E^{*}}(X)$ and $T_{E}(X, R)$ are $E$-inversive semigroups in terms of the cardinality of $X/E$ and $R$, respectively.

Full Text: PDF


  • There are currently no refbacks.

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

Copyright 2021 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|