Text this: Green’s relations and natural partial order on the regular subsemigroup of transformations preserving an equivalence relation and fixed a cross-section

 __   _      ___      ____       ___      _____   
| || | ||   / _ \\   |  _ \\    / _ \\   / ____|| 
| '--' ||  | / \ ||  | |_| ||  | / \ || / //---`' 
| .--. ||  | \_/ ||  | .  //   | \_/ || \ \\___   
|_|| |_||   \___//   |_|\_\\    \___//   \_____|| 
`-`  `-`    `---`    `-` --`    `---`     `----`