@article{Sawatraksa_Tantong_2022, place={Thailand}, title={An Observation on the Natural Partial Order of Transformation Semigroups Restricted by an Equivalence Relation}, volume={12}, url={https://ph02.tci-thaijo.org/index.php/past/article/view/246019}, DOI={10.14456/past.2022.3}, abstractNote={<p>Let  <img title="T(X)" src="https://latex.codecogs.com/gif.latex?T(X)" /> be the full transformation semigroup of a set <img title="X" src="https://latex.codecogs.com/gif.latex?X" /> . We consider the subsemigroup of  <img title="T(X)" src="https://latex.codecogs.com/gif.latex?T(X)" /> defined by <img title="E(X,\sigma)=\{\alpha\in T(X): \forall x,y\in X, (x,y)\in\sigma \;\text{implies}\; x\alpha=y\alpha\}" src="https://latex.codecogs.com/gif.latex?E(X,\sigma)=\{\alpha\in&space;T(X):&space;\forall&space;x,y\in&space;X,&space;(x,y)\in\sigma&space;\;\text{implies}\;&space;x\alpha=y\alpha\}" />  for an arbitrary equivalence relation <img title="\sigma" src="https://latex.codecogs.com/gif.latex?\sigma" />  on <img title="X" src="https://latex.codecogs.com/gif.latex?X" /> . The natural partial order on the largest regular subsemigroup of <img title="E(X,\sigma)" src="https://latex.codecogs.com/gif.latex?E(X,\sigma)" />  is discussed in this paper, and we characterize when two regular elements of  <img title="E(X,\sigma)" src="https://latex.codecogs.com/gif.latex?E(X,\sigma)" /> are related under this order. Also, their maximal, minimal and covering elements are described.</p>}, number={1}, journal={Progress in Applied Science and Technology}, author={Sawatraksa, Nares and Tantong, Piyaporn}, year={2022}, month={Apr.}, pages={17–22} }