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Regularity in the Semigroup of Transformations Preserving a Zig-Zag Order

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Abstract

It is well known that the semigroup T(X) of all transformations on a set X is regular, but its subsemigroups do not need to be. Consider an ordered set \((X;\le )\) whose order forms a path with alternating orientation. The semigroup \(\mathrm{OT}(X)\) of all order-preserving transformations on \((X;\le )\) is a subsemigroup of T(X). Some characterizations of regular semigroups \(\mathrm{OT}(X)\) were given. For a non-empty subset S of X, we define the set

$$\begin{aligned} \mathrm{OT}_S(X)=\{\alpha \in \mathrm{OT}(X)\mid {{\,\mathrm{ran}\,}}\alpha =S \}. \end{aligned}$$

In this paper, we show that in general, \(\mathrm{OT}_S(X)\) is not a subsemigroup of T(X). A necessary and sufficient condition for \(\mathrm{OT}_S(X)\) to be a regular subsemigroup of T(X) is given. Some regular subsemigroups of T(X) that are contained in \(\mathrm{OT}_S(X)\) are studied. We investigate the left (right) regularity of \(\mathrm{OT}(X)\). We obtain that left regular and right regular elements in \(\mathrm{OT}(X)\) are identical. A characterization of left (right) elements in \(\mathrm{OT}(X)\) is given. We classify all left (right) regular semigroups \(\mathrm{OT}(X)\).

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Acknowledgements

We would like to thank the referees for their thorough review and appreciate the constructive comments and suggestions, which have significantly contributed to improve the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Silpakorn University, Sanam Chan Palace Campus, Nakhon Pathom, 73000, Thailand

    Ratana Srithus & Chompunutch Khongthat

  2. Department of Mathematics, Faculty of Science, Prince of Songkla University, Hatyai, Songkhla, 90112, Thailand

    Ronnason Chinram

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  1. Ratana Srithus
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Correspondence to Ratana Srithus.

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Communicated by Kar Ping Shum.

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The research of the Ratana Srithus was supported by the Thailand Research Fund and the Higher Education Commission under Grant No. MRG6080053.

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Srithus, R., Chinram, R. & Khongthat, C. Regularity in the Semigroup of Transformations Preserving a Zig-Zag Order. Bull. Malays. Math. Sci. Soc. 43, 1761–1773 (2020). https://doi.org/10.1007/s40840-019-00772-2

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  • DOI: https://doi.org/10.1007/s40840-019-00772-2

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