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

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