Green’s relations and regularity on semigroups of partial transformations with fixed sets

Abstract

Given a non-empty set X and let P(X) be a partial transformation semigroup on X. For a fixed subset Y of X, define a generalization of P(X) by $$\mathit{PFix}(X,Y)=\{\alpha \in P(X):y\alpha =y\mathrm{ }\text{for}\mathrm{ }\text{all}\mathrm{ }y\in \text{dom}(\alpha )\cap Y\}.$$

In this article, we give a complete description of Green’s relations on PFix(X, Y) and apply this to obtain the characterizations of left regular, right regular, intra-regular and completely regular elements of PFix(X, Y). The number and the relationship between these elements are also investigated by using finiteness of X and $X\setminus Y.$

Keywords:

• Green’s relations
• intra-regularity
• left regularity
• partial transformation semigroup
• right regularity

2020 Mathematics Subject Classification:

• 20M20

Acknowledgements

We would like to thank Prof. Wutiphol Sintunavarat for a brief history of fixed point theory. We also thank the referee for his/her great efforts in proofreading the manuscript.

Additional information

Funding

This study was supported by Thammasat University Research Fund, Contract No. TUFT 71/2564.