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

  • Ratana SrithusEmail author
  • Ronnason Chinram
  • Chompunutch Khongthat
Article
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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 \(\mathrm{OT}(X)\).

Keywords

Order-preserving Fence Regular Left regular Right regular 

Mathematics Subject Classification

20M20 20M17 

Notes

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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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceSilpakorn UniversityNakhon PathomThailand
  2. 2.Department of Mathematics, Faculty of SciencePrince of Songkla UniversityHatyaiThailand

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