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A Natural Partial Order on Partition Order-Decreasing Transformation Semigroups

Original Paper
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Abstract

Mitsch defined the natural partial order \(\le \) on a semigroup S as follows: \(a\le b\) if and only if \(a=xb=by, a=xa\) for some \(x,y\in S^1\). Let \({{{\mathcal {T}}}}_X\) be the full transformation semigroup on a finite set \(X=\{1,2,\ldots ,n\}\). Let \(\rho \) be an equivalence relation on X and \(\preceq \) be a total order on the partition set \(X/\rho \) of X induced by \(\rho \). Denote by \({\overline{x}}\) the \(\rho \)-class containing \(x\in X\). In this paper, we endow the partition order-decreasing transformation subsemigroup of \({{{\mathcal {T}}}}_X\) defined by
$$\begin{aligned} T(\rho ,\preceq )=\{f\in {{{\mathcal {T}}}}_X: \forall \,\,x\in X, \overline{f(x)}\preceq {\overline{x}}\} \end{aligned}$$
with the natural partial order and give a characterization for this order. Then we determine the compatibility of their elements and find all the minimal and maximal elements.

Keywords

Transformation semigroup Natural partial order Compatibility The minimal (maximal) elements 

Mathematics Subject Classification

20M20 

Notes

Acknowledgement

I would like to thank the referee for his/her valuable suggestions and comments which helped to improve the presentation of this paper.

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Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsYancheng Institute of TechnologyYanchengChina

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