A partial order on transformation semigroups with restricted range that preserve double direction equivalence


               <jats:p>Let <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_001.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>T</m:mi>
                           <m:mrow>
                              <m:mo>(</m:mo>
                              <m:mrow>
                                 <m:mi>X</m:mi>
                              </m:mrow>
                              <m:mo>)</m:mo>
                           </m:mrow>
                        </m:math>
                        <jats:tex-math>T\left(X)</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> be the full transformation semigroup on a set <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_002.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>X</m:mi>
                        </m:math>
                        <jats:tex-math>X</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>. For an equivalence <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_003.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>E</m:mi>
                        </m:math>
                        <jats:tex-math>E</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> on <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_004.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>X</m:mi>
                        </m:math>
                        <jats:tex-math>X</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>, let <jats:disp-formula id="j_math-2021-0109_eq_001">
                     <jats:alternatives>
                        <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_005.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block">
                           <m:msub>
                              <m:mrow>
                                 <m:mi>T</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:msup>
                                    <m:mrow>
                                       <m:mi>E</m:mi>
                                    </m:mrow>
                                    <m:mrow>
                                       <m:mo>∗</m:mo>
                                    </m:mrow>
                                 </m:msup>
                              </m:mrow>
                           </m:msub>
                           <m:mrow>
                              <m:mo>(</m:mo>
                              <m:mrow>
                                 <m:mi>X</m:mi>
                              </m:mrow>
                              <m:mo>)</m:mo>
                           </m:mrow>
                           <m:mo>=</m:mo>
                           <m:mrow>
                              <m:mo>{</m:mo>
                              <m:mrow>
                                 <m:mi>α</m:mi>
                                 <m:mo>∈</m:mo>
                                 <m:mi>T</m:mi>
                                 <m:mrow>
                                    <m:mo>(</m:mo>
                                    <m:mrow>
                                       <m:mi>X</m:mi>
                                    </m:mrow>
                                    <m:mo>)</m:mo>
                                 </m:mrow>
                                 <m:mo>:</m:mo>
                                 <m:mrow>
                                    <m:mo>∀</m:mo>
                                 </m:mrow>
                                 <m:mi>x</m:mi>
                                 <m:mo>,</m:mo>
                                 <m:mi>y</m:mi>
                                 <m:mo>∈</m:mo>
                                 <m:mi>X</m:mi>
                                 <m:mo>,</m:mo>
                                 <m:mrow>
                                    <m:mo>(</m:mo>
                                    <m:mrow>
                                       <m:mi>x</m:mi>
                                       <m:mo>,</m:mo>
                                       <m:mi>y</m:mi>
                                    </m:mrow>
                                    <m:mo>)</m:mo>
                                 </m:mrow>
                                 <m:mo>∈</m:mo>
                                 <m:mi>E</m:mi>
                                 <m:mo>⇔</m:mo>
                                 <m:mrow>
                                    <m:mo>(</m:mo>
                                    <m:mrow>
                                       <m:mi>x</m:mi>
                                       <m:mi>α</m:mi>
                                       <m:mo>,</m:mo>
                                       <m:mi>y</m:mi>
                                       <m:mi>α</m:mi>
                                    </m:mrow>
                                    <m:mo>)</m:mo>
                                 </m:mrow>
                                 <m:mo>∈</m:mo>
                                 <m:mi>E</m:mi>
                              </m:mrow>
                              <m:mo>}</m:mo>
                           </m:mrow>
                           <m:mo>.</m:mo>
                        </m:math>
                        <jats:tex-math>{T}_{{E}^{\ast }}\left(X)=\left\{\alpha \in T\left(X):\forall x,y\in X,\left(x,y)\in E\iff \left(x\alpha ,y\alpha )\in E\right\}.</jats:tex-math>
                     </jats:alternatives>
                  </jats:disp-formula> For each nonempty subset <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_006.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>Y</m:mi>
                        </m:math>
                        <jats:tex-math>Y</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> of <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_007.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>X</m:mi>
                        </m:math>
                        <jats:tex-math>X</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>, we denote the restriction of <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_008.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>E</m:mi>
                        </m:math>
                        <jats:tex-math>E</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> to <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_009.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>Y</m:mi>
                        </m:math>
                        <jats:tex-math>Y</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> by <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_010.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:msub>
                              <m:mrow>
                                 <m:mi>E</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:mi>Y</m:mi>
                              </m:mrow>
                           </m:msub>
                        </m:math>
                        <jats:tex-math>{E}_{Y}</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>. Let <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_011.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:msub>
                              <m:mrow>
                                 <m:mi>T</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:msup>
                                    <m:mrow>
                                       <m:mi>E</m:mi>
                                    </m:mrow>
                                    <m:mrow>
                                       <m:mo>∗</m:mo>
                                    </m:mrow>
                                 </m:msup>
                              </m:mrow>
                           </m:msub>
                           <m:mrow>
                              <m:mo>(</m:mo>
                              <m:mrow>
                                 <m:mi>X</m:mi>
                                 <m:mo>,</m:mo>
                                 <m:mi>Y</m:mi>
                              </m:mrow>
                              <m:mo>)</m:mo>
                           </m:mrow>
                        </m:math>
                        <jats:tex-math>{T}_{{E}^{\ast }}\left(X,Y)</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> be the intersection of the semigroup <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_012.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:msub>
                              <m:mrow>
                                 <m:mi>T</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:msup>
                                    <m:mrow>
                                       <m:mi>E</m:mi>
                                    </m:mrow>
                                    <m:mrow>
                                       <m:mo>∗</m:mo>
                                    </m:mrow>
                                 </m:msup>
                              </m:mrow>
                           </m:msub>
                           <m:mrow>
                              <m:mo>(</m:mo>
                              <m:mrow>
                                 <m:mi>X</m:mi>
                              </m:mrow>
                              <m:mo>)</m:mo>
                           </m:mrow>
                        </m:math>
                        <jats:tex-math>{T}_{{E}^{\ast }}\left(X)</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> with the semigroup of all transformations with restricted range <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_013.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>Y</m:mi>
                        </m:math>
                        <jats:tex-math>Y</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> under the condition that <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_014.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mo>∣</m:mo>
                           <m:mi>X</m:mi>
                           <m:mspace width="-0.1em" />
                           <m:mtext>/</m:mtext>
                           <m:mi>E</m:mi>
                           <m:mo>∣</m:mo>
                           <m:mo>=</m:mo>
                           <m:mo>∣</m:mo>
                           <m:mi>Y</m:mi>
                           <m:mspace width="-0.16em" />
                           <m:mtext>/</m:mtext>
                           <m:mspace width="-0.1em" />
                           <m:msub>
                              <m:mrow>
                                 <m:mi>E</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:mi>Y</m:mi>
                              </m:mrow>
                           </m:msub>
                           <m:mo>∣</m:mo>
                        </m:math>
                        <jats:tex-math>| X\hspace{-0.1em}\text{/}E| =| Y\hspace{-0.16em}\text{/}\hspace{-0.1em}{E}_{Y}| </jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>. Equivalently, <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_015.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:msub>
                              <m:mrow>
                                 <m:mi>T</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:msup>
                                    <m:mrow>
                                       <m:mi>E</m:mi>
                                    </m:mrow>
                                    <m:mrow>
                                       <m:mo>∗</m:mo>
                                    </m:mrow>
                                 </m:msup>
                              </m:mrow>
                           </m:msub>
                           <m:mrow>
                              <m:mo>(</m:mo>
                              <m:mrow>
                                 <m:mi>X</m:mi>
                                 <m:mo>,</m:mo>
                                 <m:mi>Y</m:mi>
                              </m:mrow>
                              <m:mo>)</m:mo>
                           </m:mrow>
                           <m:mo>=</m:mo>
                           <m:mrow>
                              <m:mo>{</m:mo>
                              <m:mrow>
                                 <m:mi>α</m:mi>
                                 <m:mo>∈</m:mo>
                                 <m:msub>
                                    <m:mrow>
                                       <m:mi>T</m:mi>
                                    </m:mrow>
                                    <m:mrow>
                                       <m:msup>
                                          <m:mrow>
                                             <m:mi>E</m:mi>
                                          </m:mrow>
                                          <m:mrow>
                                             <m:mo>∗</m:mo>
                                          </m:mrow>
                                       </m:msup>
                                    </m:mrow>
                                 </m:msub>
                                 <m:mrow>
                                    <m:mo>(</m:mo>
                                    <m:mrow>
                                       <m:mi>X</m:mi>
                                    </m:mrow>
                                    <m:mo>)</m:mo>
                                 </m:mrow>
                                 <m:mo>:</m:mo>
                                 <m:mi>X</m:mi>
                                 <m:mi>α</m:mi>
                                 <m:mo>⊆</m:mo>
                                 <m:mi>Y</m:mi>
                              </m:mrow>
                              <m:mo>}</m:mo>
                           </m:mrow>
                        </m:math>
                        <jats:tex-math>{T}_{{E}^{\ast }}\left(X,Y)=\left\{\alpha \in {T}_{{E}^{\ast }}\left(X):X\alpha \subseteq Y\right\}</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>, where <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_016.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mo>∣</m:mo>
                           <m:mi>X</m:mi>
                           <m:mspace width="-0.1em" />
                           <m:mtext>/</m:mtext>
                           <m:mspace width="-0.1em" />
                           <m:mi>E</m:mi>
                           <m:mo>∣</m:mo>
                           <m:mo>=</m:mo>
                           <m:mo>∣</m:mo>
                           <m:mi>Y</m:mi>
                           <m:mspace width="-0.16em" />
                           <m:mtext>/</m:mtext>
                           <m:mspace width="-0.1em" />
                           <m:msub>
                              <m:mrow>
                                 <m:mi>E</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:mi>Y</m:mi>
                              </m:mrow>
                           </m:msub>
                           <m:mo>∣</m:mo>
                        </m:math>
                        <jats:tex-math>| X\hspace{-0.1em}\text{/}\hspace{-0.1em}E| =| Y\hspace{-0.16em}\text{/}\hspace{-0.1em}{E}_{Y}| </jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>. Then <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_017.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:msub>
                              <m:mrow>
                                 <m:mi>T</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:msup>
                                    <m:mrow>
                                       <m:mi>E</m:mi>
                                    </m:mrow>
                                    <m:mrow>
                                       <m:mo>∗</m:mo>
                                    </m:mrow>
                                 </m:msup>
                              </m:mrow>
                           </m:msub>
                           <m:mrow>
                              <m:mo>(</m:mo>
                              <m:mrow>
                                 <m:mi>X</m:mi>
                                 <m:mo>,</m:mo>
                                 <m:mi>Y</m:mi>
                              </m:mrow>
                              <m:mo>)</m:mo>
                           </m:mrow>
                        </m:math>
                        <jats:tex-math>{T}_{{E}^{\ast }}\left(X,Y)</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> is a subsemigroup of <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_018.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:msub>
                              <m:mrow>
                                 <m:mi>T</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:msup>
                                    <m:mrow>
                                       <m:mi>E</m:mi>
                                    </m:mrow>
                                    <m:mrow>
                                       <m:mo>∗</m:mo>
                                    </m:mrow>
                                 </m:msup>
                              </m:mrow>
                           </m:msub>
                           <m:mrow>
                              <m:mo>(</m:mo>
                              <m:mrow>
                                 <m:mi>X</m:mi>
                              </m:mrow>
                              <m:mo>)</m:mo>
                           </m:mrow>
                        </m:math>
                        <jats:tex-math>{T}_{{E}^{\ast }}\left(X)</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>. In this paper, we characterize the natural partial order on <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_019.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:msub>
                              <m:mrow>
                                 <m:mi>T</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:msup>
                                    <m:mrow>
                                       <m:mi>E</m:mi>
                                    </m:mrow>
                                    <m:mrow>
                                       <m:mo>∗</m:mo>
                                    </m:mrow>
                                 </m:msup>
                              </m:mrow>
                           </m:msub>
                           <m:mrow>
                              <m:mo>(</m:mo>
                              <m:mrow>
                                 <m:mi>X</m:mi>
                                 <m:mo>,</m:mo>
                                 <m:mi>Y</m:mi>
                              </m:mrow>
                              <m:mo>)</m:mo>
                           </m:mrow>
                        </m:math>
                        <jats:tex-math>{T}_{{E}^{\ast }}\left(X,Y)</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>. Then we find the elements which are compatible and describe the maximal and minimal elements. We also prove that every element of <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0109_eq_020.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:msub>
                              <m:mrow>
                                 <m:mi>T</m:mi>
                              </m:mrow>
                              <m:mrow>
                                 <m:msup>
                                    <m:mrow>
                                       <m:mi>E</m:mi>
                                    </m:mrow>
                                    <m:mrow>
                                       <m:mo>∗</m:mo>
                                    </m:mrow>
                                 </m:msup>
                              </m:mrow>
                           </m:msub>
                           <m:mrow>
                              <m:mo>(</m:mo>
                              <m:mrow>
                                 <m:mi>X</m:mi>
                                 <m:mo>,</m:mo>
                                 <m:mi>Y</m:mi>
                              </m:mrow>
                              <m:mo>)</m:mo>
                           </m:mrow>
                        </m:math>
                        <jats:tex-math>{T}_{{E}^{\ast }}\left(X,Y)</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> lies between maximal and minimal elements. Finally, the existence of an upper cover and a lower cover is investigated.</jats:p>

In this paper, we characterize the natural partial order on T X Y , E ( ) * . Then we find the elements which are compatible and describe the maximal and minimal elements. We also prove that every element of T X Y , E ( )

Introduction and preliminaries
Let T X ( ) be the set of all functions from a set X into itself. Then T X ( ) under the composition of functions is a semigroup which is called the full transformation semigroup on X. In 1975, Symons [1] studied the semigroup T X Y , ( ) defined by where Y is a nonempty subset of X. Note that if we let α T X ( ) ∈ and Z X ⊆ , the notation Zα means zα z Z : In [1], the author studied the automorphism of T X Y , ( )and the isomorphism between two semigroups T X Y , 1 1 ( ) and T X Y , 2 2 ( ). In 2008, Sanwong and Sommanee [2] found a necessary and sufficient condition for T X Y , ( )to be regular. Moreover, the largest regular subsemigroup was obtained. Later on, Sangkhanan and Sanwong [3] and, independently, Sun and Sun [4] endowed T X Y , ( ) with ≤, the natural partial order, and determined when two elements of T X Y , ( )related under this order, then found out elements of T X Y , ( )which are compatible with ≤ on T X Y , ( ). They also described its maximal and minimal elements and proved that every element in T X Y , ( )must lie between maximal and minimal elements. Let E be an equivalence on X. In 2005, Pei [5] defined a subsemigroup T X E ( ) of T X ( ) by T X α T X x y X x y E xα yα E : , , , , .
Then T X E ( ) is exactly S X ( ), the semigroup of all continuous self-maps of the topological space X for which all E-classes form a basis. In [5], the author investigated regularity of elements and Green's relations for T X E ( ). In 2008, Sun et al. [6] described T X E ( ) with the natural partial order and gave a characterization for two elements of T X E ( ) related under this order. They found out elements of T X E ( ) which are compatible with ≤ on T X E ( ). In addition, the maximal, minimal and covering elements were described. Recently, Sangkhanan and Sanwong [7] defined a subsemigroup T X Y , )is the semigroup of all continuous self-maps of the topological space X for which all E-classes form a basis carrying X into a subspace Y . In [7], they gave a necessary and sufficient condition for T X Y , E ( ) to be regular and characterized Green's relations on T X Y , E ( ). In 2010, Deng et al. [8] introduced a subsemigroup T X E ( ) * of T X ( ) by Then T X E ( ) * is a semigroup of continuous self-maps of the topological space X for which all E-classes form a basis. The authors studied regularity of elements and Green's relations for T X E ( ) * . In 2013, Sun and Sun [9] endowed T X E ( ) * with the natural partial order. They determined when two elements are related under this order and found the elements which are compatible. Moreover, the maximal and minimal elements were described. Finally, they studied the existence of the greatest lower bound of two elements.
is a semigroup of continuous self-maps of the topological space X for which all E classes form a basis carrying X into a subspace Y , and is referred to as a semigroup of continuous functions (see [11] for details).
In this paper, we aim to generalize the results of [3,4,9]. Actually, we study the natural partial order on which extends the results on T X E ( ) * and T X Y , ( ) and improve some results in these semigroups. For example, in Section 4, we characterize the left and right compatible elements instead of the strictly compatibility which was studied in [9]. Moreover, in Section 5, we show that every element in T X Y , E ( ) * lies between maximal and minimal elements which have never been studied before in T X E ( ) * . We first state some notations and results that will be used later. For each α T X Y , E ( ) ∈ * and A X ⊆ , the restriction of α to A is denoted by α A | . We adopt the notation introduced in [ , the symbol π α ( ) will denote the decomposition of X induced by the map α, namely, where α x y X X xα yα ker , : For a subset A of X, we put π α M π α M A : We define the restriction of the equivalence E on a subset M of X by where the notation S 1 denotes a monoid obtained from S by adjoining an identity 1 if necessary (S S 1 = for a monoid S). In this paper, we use (1) to define the partial order on the semigroup T X Y , has no identity elements. Hence, in this case, T X Y T X Y , , . It is worth studying the natural partial order on T X Y , Let us refer to the following corollary which will prove useful.
and α β ≤ . Then the following statements hold.

Characterization
Now, we start this section with the characterization of ≤ on T X Y , E ( ) * , which extends Theorem 2.1 of [9].
The proof is an appropriate modification of the proof of Theorem 2.1 of [9]. In fact, assume that (1), (2) and (3) hold. Note that for each A X E ∈ / , Aα is nonempty. Hence, by By using the same method as in the proof of Theorem 2.1 in [9], we can see We can see that Theorem 2.1 becomes Theorem 2.1 of [9].
Proof. Let α β ≤ and suppose that α β < . Then for is nonempty and so A Y ∩ is also nonempty for all classes A. It leads to a contradiction. Therefore, α β = . □ By the above proposition, we can see that if Y A ∩ = ∅for some class A X E ∈ / , then the natural partial order ≤ becomes the equality relation. From now on, we assume that Y A ∩ is nonempty for all E-classes A,

Compatibility
Recall that an element γ T X Y , E ( ) ∈ * is said to be strictly left compatible if γα γβ < whenever α β < . Strictly right compatibility is defined dually. We note that, in Section 2 of [9], the authors studied the strictly compatibility on T X E ( ) * but, in this paper, we remove the term "strictly," that is, Right compatibility is defined dually.
Proof. Assume that γ is right compatible and suppose to the contrary that there is It is straightforward to show that α β < . We can see that sαγ sγ w = = = tγ tβγ = , which implies that tβγ Xαγ ∈ but tαγ uγ v w tγ tβγ = = ≠ = = . Thus, αγ βγ ≰ , which contradicts to the right compatibility of γ.
We show that αγ βγ ≤ by using Theorem 2.1.
We consider the following two cases.
By the above two theorems, we obtain the following two corollaries which appear in [3,4,9].

Maximal and minimal elements
In this section, we characterize maximal and minimal elements and show that every element of T X Y , E ( ) * must lie between maximal and minimal elements. Furthermore, we obtain the existence of an upper cover and a lower cover. Sanwong and Sommanee and proved that F is the largest regular subsemigroup of T X Y , ( ). Moreover, in [10], the authors defined the set F F T X Y , : .
(3) It is clear that Cα , there is y A Y ∈ ∩ such that aα yα = . If y p = , then aα pα qα qβ Hence, x Xα Yα ∈ = , which implies that x yα = for some y Y ∈ and so y xα We can see that C d C { } ⧹ is nonempty since C 1 | | > . Consider the following two cases.
Case 1: Then there is an injection .
Then there is an injection .
Clearly, β i is injective. Finally, we define a function β X Y : → by It is easy to see that β T X Y , E ( ) ∈ * and α β ≠ . Moreover, β satisfies Theorem 4.2, which implies that β is maximal. Now, we show that α β ≤ by using Theorem 2.1.
(2) By the definition of β, it is easy to see that for each x X We consider the following two cases.
For A i satisfies case 1 as above, it is clear that xα xβ . We can see that xα d α . We can see that xα d α The proof of the above theorem can be illustrated by the following example.   Now, we deal with minimal elements.
. Then α is minimal if and only if Aα Proof. (⇒) We prove by contrapositive. Assume that Aα It is easy to verify that β T X Y , E ( ) ∈ * and β α ≠ . We show that β α ≤ .
Proof. Assume that α is not minimal. Then Aα 1 | | > for some A X E ∈ / . Let β be defined as in the proof of Theorem 4.5. Then β α < . Moreover, it is clear that Cβ 1 | | = for each C X E ∈ / . Hence, β is minimal. □ By the above results, we may conclude the following corollaries which extend the results in [3,4,9].

α is maximal if and only if for each E-class A, Aα X E
∈ / or α A | is injective.

α is minimal if and only if Aα
There exists a maximal element β T X E ( ) ∈ * such that α β ≤ . 4. There exists a minimal element β T X E ( ) ∈ * such that β α ≤ .
. Then the following statements hold.
Finally, the following results are concerned with the existence of an upper cover and a lower cover for Recall that an element b in a semigroup S is called an upper cover for a S ∈ if a b < and there exists no c S ∈ such that a c b < < . A lower cover is defined dually.
. If α is not maximal, then α has an upper cover. . If α is not minimal, then α has a lower cover.
We end this section by briefly summarizing our paper and point out future work. In [3,4,9], the natural partial order on T X Y , ( )and T X E ( ) * was determined. This paper is concerned with the same problems for the semigroup T X Y , E ( ) * , that is, we study the compatibility, maximality and minimality of its elements in this semigroup under the natural partial order. Moreover, in the last section of our paper, we show that every element of T X Y , E ( ) * lies between maximal and minimal elements. These are results that we have not found in the studies of other transformation semigroups. We also prove that for every α T X Y , E ( ) ∈ * , if α is not maximal, then it has an upper cover; and if α is not minimal, then it has a lower cover. However, in [4], the authors investigated the greatest lower bound and the least upper bound of two elements of T X E ( ) * , it is then natural to ask for this property on T X Y , E ( ) * , which is an open question.