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Regularity and Green’s relations on semigroups of transformations with restricted range that preserve an equivalence

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Abstract

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

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

Then \(T_E(X,Y)\) is the semigroup of all continuous self-maps of the topological space X for which all E-classes form a basis carrying X into a subspace Y. In this paper, we give a necessary and sufficient condition for \(T_E(X,Y)\) to be regular and characterize Green’s relations on \(T_E(X,Y)\). Our work extends previous results found in the literature.

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Acknowledgements

This research was supported by the Thailand Research Fund under Grant No. TRG5880113 and Chiang Mai University. The authors would like to thank the referee for the careful review, which improves the readability of this paper. The first author thanks Teerapong Suksumran for his collaboration.

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Correspondence to Kritsada Sangkhanan.

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Communicated by Marcel Jackson.

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Sangkhanan, K., Sanwong, J. Regularity and Green’s relations on semigroups of transformations with restricted range that preserve an equivalence. Semigroup Forum 100, 568–584 (2020). https://doi.org/10.1007/s00233-020-10089-3

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  • DOI: https://doi.org/10.1007/s00233-020-10089-3

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