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

• Ratana Srithus
• Ronnason Chinram
• Chompunutch Khongthat
Article

## 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

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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© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

## Authors and Affiliations

• Ratana Srithus
• 1
• Ronnason Chinram
• 2
• Chompunutch Khongthat
• 1
1. 1.Department of Mathematics, Faculty of ScienceSilpakorn UniversityNakhon PathomThailand
2. 2.Department of Mathematics, Faculty of SciencePrince of Songkla UniversityHatyaiThailand