A Natural Partial Order on Partition Order-Decreasing Transformation Semigroups


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.

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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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Sun, L. A Natural Partial Order on Partition Order-Decreasing Transformation Semigroups. Bull. Iran. Math. Soc. 46, 1357–1369 (2020). https://doi.org/10.1007/s41980-019-00329-w

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

Mathematics Subject Classification

  • 20M20