# Natural Partial Order on Semigroups of Transformations Preserving an Equivalence Relation and a Cross-Section with Fixed Sets

## Abstract

Let $\small&space;T(X,\rho,R)$ be the semigroup consisting of all total transformations preserving an equivalence relation $\small&space;\rho$ and a cross-section $R$. The subsemigroup $T_{Y}(X,\rho,R)$  of $T(X,\rho,R)$ is defined as follows:

$\small&space;T_Y(X,\rho,R)=\{\alpha\in&space;T(X,\rho,R)&space;:&space;x\alpha=x~\text{for&space;all}~x\in&space;X~\text{in&space;which}~x\rho\cap&space;Y\neq\emptyset\}$.

In this paper, we characterize the natural partial order and find elements which are left compatible, right compatible and compatible.

Section
บทความวิจัย

## References

Araújo, J. & Konieczny, J. (2003). Automorphism groups of centralizers of idempotents. Journal of Algebra, 269, 227-239. https://doi.org/10.1016/S0021-8693(03)00499-X

Araújo, J. & Konieczny, J. (2004). Semigroups of transformations preserving an equivalence relation and a cross-section. Communications in Algebra, 32, 1917-1935. https://doi.org/10.1081/AGB-120029913

Chaiya, Y., Honyam, P. & Sanwong, J. (2016). Natural partial orders on transformation semigroups with fixed sets. International Journal of Mathematics and Mathematical Sciences, 2016, https://doi.org/10.1155/2016/2759090

Clifford, A.H. & Preston, G.B. (1961). The Algebraic Theory of Semigroups. American Mathematical Society, USA.

Hartwig, R.E. (1980). How to partially order regular elements. Mathematica Japonica, 25(1), 1-13.

Honyam, P. & Sanwong, J. (2013). Semigroups of transformations with fixed sets. Quaestiones Mathematicae, 36, 79-92. https://doi.org/10.2989/16073606.2013.779958

Howie, J.M. (1995). Fundamentals of Semigroup Theory. Oxford University Press, New York.

Pei, H. & Deng, W. (2013). Naturally ordered semigroups of partial transformations preserving an equivalence relation. Communications in Algebra, 41, 3308-3324. https://doi.org/10.1080/00927872.2012.684083

Kowol, G. & Mitsch, H. (1986). Naturally ordered transformation semigroups. Monatshefte für Mathematik, 102, 115-138.

Mitsch, H. (1986). A natural partial order for semigroups. Proceedings of the American Mathe-matical Society, 97(3), 384-388.

Nambooripad, K.S.S. (1980). The natural partial order on a regular semigroup. Proceedings of the Edinburgh Mathematical Society, 23(3), 249-260.

Nupo, N. & Pookpienlert, C. (2021). Regularity on semigroups of transformations preserving an equivalence relation and a cross-section with fixed sets. International Journal of Mathematics and Computer Science, 16(1), 119-132.

Sawatraksa, N., Namnak, C. & Sangkhanan, K. (2019). Green’s relations and natural partial order on the regular subsemigroup of transformations preserving an equivalence relation and fixed a cross-section. Thai Journal of Mathematics, 17(2), 431-444.

Sun, L. & Sun, J. (2013). A partial order on transformation semigroups that preserve double direction equivalence relation. Journal of Algebra and Its Applications, 12(8), https://doi.org/10.1142/S0219498813500412

Sun, L., Deng, W. & Pei, H. (2011). Naturally ordered transformation semigroups preserving an equivalence relation and a cross-section. Algebra Colloquium, 18(3), 523-532. https://doi.org/10.1142/S100538671100040X