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The natural partial order on semigroups of transformations with restricted range that preserve an equivalence

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Abstract

Let Y be a nonempty subset of X and T(X, Y) the set of all functions from X into Y. Then T(X, Y) with composition is a subsemigroup of the full transformation semigroup T(X). Let E be a nontrivial equivalence on X. Define a subsemigroup \(T_E(X,Y)\) of T(X, Y) by

$$\begin{aligned} T_E(X,Y)=\{\alpha \in T(X,Y):\forall (x,y)\in E, (x\alpha ,y\alpha )\in E\}. \end{aligned}$$

We study \(T_E(X,Y)\) with the natural partial order and determine when two elements are related under this order. We also give a characterization of compatibility on \(T_E(X,Y)\) and then describe the maximal and minimal elements.

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Acknowledgements

This research was supported by the Thailand Research Fund under grant No. TRG5880113 and Chiang Mai University.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Kritsada Sangkhanan & Jintana Sanwong

Authors
  1. Kritsada Sangkhanan
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  2. Jintana Sanwong
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Correspondence to Kritsada Sangkhanan.

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Communicated by Marcel Jackson.

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Part of this work was presented in the Arbeitstagung Allgemeine Algebra conference (AAA96), Darmstadt, Germany, June 1–3, 2018

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Sangkhanan, K., Sanwong, J. The natural partial order on semigroups of transformations with restricted range that preserve an equivalence. Semigroup Forum (2024). https://doi.org/10.1007/s00233-024-10422-0

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