Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-02-11T15:22:21.879Z Has data issue: false hasContentIssue false
  • English
  • Français

THE NATURAL PARTIAL ORDER ON THE SEMIGROUP OF ALL TRANSFORMATIONS OF A SET THAT REFLECT AN EQUIVALENCE RELATION

Part of: Semigroups

Published online by Cambridge University Press:  25 January 2013

LEI SUN  and
XIANGJUN XIN
LEI SUN*
Affiliation:
School of Mathematics and Information Science, Henan Polytechnic University, Henan, Jiaozuo 454003, PR China
XIANGJUN XIN
Affiliation:
Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Henan, Zhengzhou 450002, PR China email xin_xiang_jun@126.com
*
✉sunlei97@163.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ${ \mathcal{T} }_{X} $ be the full transformation semigroup on a set $X$ and $E$ be a nontrivial equivalence relation on $X$. Denote

$$\begin{eqnarray*}{T}_{\exists } (X)= \{ f\in { \mathcal{T} }_{X} : \forall x, y\in X, (f(x), f(y))\in E\Rightarrow (x, y)\in E\} ,\end{eqnarray*}$$
so that ${T}_{\exists } (X)$ is a subsemigroup of ${ \mathcal{T} }_{X} $. In this paper, we endow ${T}_{\exists } (X)$ with the natural partial order and investigate when two elements are related, then find elements which are compatible. Also, we characterise the minimal and maximal elements.

Keywords

transformation semigroupnatural partial ordercompatibilityminimal (maximal) elements

MSC classification

Secondary: 20M20: Semigroups of transformations, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 3 , December 2013 , pp. 359 - 368
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Deng, L. Z., Zeng, J. W. and You, T. J., ‘Green’s relations and regularity for semigroups of transformations that preserve reverse direction equivalence’, Semigroup Forum 83 (2011), 489–498.CrossRefGoogle Scholar
Kowol, G. and Mitsch, H., ‘Naturally ordered transformation semigroups’, Monatsh. Math. 102 (1986), 115–138.CrossRefGoogle Scholar
Marques-Smith, M. P. O. and Sullivan, R. P., ‘Partial orders on transformation semigroups’, Monatsh. Math. 140 (2003), 103–118.Google Scholar
Mitsch, H., ‘A natural partial order for semigroups’, Proc. Amer. Math. Soc. 97 (1986), 384–388.Google Scholar
Pei, H. S., ‘Equivalences, $\alpha $-semigroups and $\alpha $-congruences’, Semigroup Forum 49 (1994), 49–58.Google Scholar
Pei, H. S., ‘A regular $\alpha $-semigroup inducing a certain lattice’, Semigroup Forum 53 (1996), 98–113.Google Scholar
Pei, H. S., ‘Some $\alpha $-semigroups inducing certain lattices’, Semigroup Forum 57 (1998), 48–59.Google Scholar
Pei, H. S., ‘On the rank of the semigroup ${T}_{E} (X)$’, Semigroup Forum 70 (2005), 107–117.Google Scholar
Pei, H. S., ‘Regularity and Green’s relations for semigroups of transformations that preserve an equivalence’, Comm. Algebra 33 (2005), 109–118.Google Scholar
Singha, B., Sanwong, J. and Sullivan, R. P., ‘Partial order on partial Baer–Levi semigroups’, Bull. Aust. Math. Soc. 81 (2010), 197–206.Google Scholar
Sullivan, R. P., ‘Partial orders on linear transformation semigroups’, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), 413–437.CrossRefGoogle Scholar
Sun, L., Pei, H. S. and Cheng, Z. X., ‘Naturally ordered transformation semigroups preserving an equivalence’, Bull. Aust. Math. Soc. 78 (2008), 117–128.CrossRefGoogle Scholar
Sun, L. and Wang, L. M., ‘Natural partial order in semigroups of transformations with invariant set’, Bull. Aust. Math. Soc. 87 (2013), 94–107.CrossRefGoogle Scholar