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<a id="X80C6C718801855E9" name="X80C6C718801855E9"></a>
<div class="ChapSects"><a href="chap10.html#X80C6C718801855E9">10 
 Green's relations
 </a>
<div class="ContSect"> <a href="chap10.html#X788D6753849BAD7C">10.1 
 Creating Green's classes and representatives
 </a>

<div class="ContSSBlock">
 <a href="chap10.html#X87558FEF805D24E1">10.1-1 XClassOfYClass</a>

 <a href="chap10.html#X81B7AD4C7C552867">10.1-2 GreensXClassOfElement</a>

 <a href="chap10.html#X7B44317786571F8B">10.1-3 GreensXClassOfElementNC</a>

 <a href="chap10.html#X7D51218A80234DE5">10.1-4 GreensXClasses</a>

 <a href="chap10.html#X865387A87FAAC395">10.1-5 XClassReps</a>

 <a href="chap10.html#X81E5A04F7DA3A1E1">10.1-6 MinimalDClass</a>
 <a href="chap10.html#X834172F4787A565B">10.1-7 MaximalXClasses</a>

 <a href="chap10.html#X7AA3F0A77D0043FB">10.1-8 NrRegularDClasses</a>
 <a href="chap10.html#X7E45FD9F7BADDFBD">10.1-9 NrXClasses</a>

 <a href="chap10.html#X8140814084748101">10.1-10 PartialOrderOfXClasses</a>

 <a href="chap10.html#X83B0EDA57F1D2F97">10.1-11 LengthOfLongestDClassChain</a>
 <a href="chap10.html#X7E872C5381D0DD8A">10.1-12 IsGreensDGreaterThanFunc</a>
</div></div>
<div class="ContSect"> <a href="chap10.html#X819CCBD67FD27115">10.2 
 Iterators and enumerators of classes and representatives
 </a>

<div class="ContSSBlock">
 <a href="chap10.html#X8566F84A7F6D4193">10.2-1 IteratorOfXClassReps</a>

 <a href="chap10.html#X867D7B8982915960">10.2-2 IteratorOfXClasses</a>

</div></div>
<div class="ContSect"> <a href="chap10.html#X820EF2BA7D5D53B4">10.3 
 Properties of Green's classes
 </a>

<div class="ContSSBlock">
 <a href="chap10.html#X85F30ACF86C3A733">10.3-1 Less than for Green's classes</a>

 <a href="chap10.html#X859DD1C079C80DCC">10.3-2 IsRegularGreensClass</a>
 <a href="chap10.html#X7E9BD34B8021045A">10.3-3 IsGreensClassNC</a>
</div></div>
<div class="ContSect"> <a href="chap10.html#X855723B17D4AAF8F">10.4 
 Attributes of Green's classes
 </a>

<div class="ContSSBlock">
 <a href="chap10.html#X8723756387DD4C0F">10.4-1 GroupHClass</a>
 <a href="chap10.html#X84F1321E8217D2A8">10.4-2 SchutzenbergerGroup</a>
 <a href="chap10.html#X81202126806443F9">10.4-3 StructureDescriptionSchutzenbergerGroups</a>
 <a href="chap10.html#X838F43FE79A8C678">10.4-4 StructureDescriptionMaximalSubgroups</a>
 <a href="chap10.html#X8459E4067C5773AD">10.4-5 MultiplicativeNeutralElement</a>
 <a href="chap10.html#X85B34FFB82C83127">10.4-6 StructureDescription</a>
 <a href="chap10.html#X7EBB4F1981CC2AE9">10.4-7 InjectionPrincipalFactor</a>
 <a href="chap10.html#X86C6D777847AAEC7">10.4-8 PrincipalFactor</a>
</div></div>
<div class="ContSect"> <a href="chap10.html#X802E2BC9828341A2">10.5 
 Operations for Green's relations and classes
 </a>

<div class="ContSSBlock">
 <a href="chap10.html#X7EDE3F03879B2B12">10.5-1 LeftGreensMultiplier</a>
</div></div>
</div>

<h3>10 
 Green's relations
 </h3>

In this chapter we describe the functions in Semigroups for computing Green's classes and related properties of semigroups.

<a id="X788D6753849BAD7C" name="X788D6753849BAD7C"></a>

<h4>10.1 
 Creating Green's classes and representatives
 </h4>

In this section, we describe the methods in the Semigroups package for creating Green's classes.

<a id="X87558FEF805D24E1" name="X87558FEF805D24E1"></a>

<h5>10.1-1 XClassOfYClass</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DClassOfHClass</code>( <var class="Arg">class</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DClassOfLClass</code>( <var class="Arg">class</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DClassOfRClass</code>( <var class="Arg">class</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LClassOfHClass</code>( <var class="Arg">class</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RClassOfHClass</code>( <var class="Arg">class</var> )</td><td class="tdright">( method )</td></tr></table></div>
Returns: A Green's class.

<code class="code">XClassOfYClass</code> returns the <code class="code">X</code>-class containing the <code class="code">Y</code>-class <var class="Arg">class</var> where <code class="code">X</code> and <code class="code">Y</code> should be replaced by an appropriate choice of <code class="code">D, H, L,</code> and <code class="code">R</code>.

Note that if it is not known to GAP whether or not the representative of <var class="Arg">class</var> is an element of the semigroup containing <var class="Arg">class</var>, then no attempt is made to check this.

The same result can be produced using:

<div class="example"><pre>First(GreensXClasses(S), x -> Representative(x) in class);</pre></div>

but this might be substantially slower. Note that <code class="code">XClassOfYClass</code> is also likely to be faster than

<div class="example"><pre>GreensXClassOfElement(S, Representative(class));</pre></div>

<code class="code">DClass</code> can also be used as a synonym for <code class="code">DClassOfHClass</code>, <code class="code">DClassOfLClass</code>, and <code class="code">DClassOfRClass</code>; <code class="code">LClass</code> as a synonym for <code class="code">LClassOfHClass</code>; and <code class="code">RClass</code> as a synonym for <code class="code">RClassOfHClass</code>. See also <code class="func">GreensDClassOfElement</code> (<a href="../../../doc/ref/chap51_mj.html#X87C75A9D86122D93">Reference: GreensDClassOfElement</a>) and <code class="func">GreensDClassOfElementNC</code> (<a href="chap10.html#X7B44317786571F8B">10.1-3</a>).

<div class="example"><pre>
gap> S := Semigroup(Transformation([1, 3, 2]),
> Transformation([2, 1, 3]),
> Transformation([3, 2, 1]),
> Transformation([1, 3, 1]));;
gap> R := GreensRClassOfElement(S, Transformation([3, 2, 1]));
<Green's R-class: Transformation( [ 3, 2, 1 ] )>
gap> DClassOfRClass(R);
<Green's D-class: Transformation( [ 3, 2, 1 ] )>
gap> IsGreensDClass(DClassOfRClass(R));
true
gap> S := InverseSemigroup(
> PartialPerm([2, 6, 7, 0, 0, 9, 0, 1, 0, 5]),
> PartialPerm([3, 8, 1, 9, 0, 4, 10, 5, 0, 6]));
<inverse partial perm semigroup of rank 10 with 2 generators>
gap> x := S.1;
[3,7][8,1,2,6,9][10,5]
gap> H := HClass(S, x);
<Green's H-class: [3,7][8,1,2,6,9][10,5]>
gap> R := RClassOfHClass(H);
<Green's R-class: [3,7][8,1,2,6,9][10,5]>
gap> L := LClass(H);;
gap> L = LClass(S, PartialPerm([1, 2, 0, 0, 5, 6, 7, 0, 9]));
true
gap> DClass(R) = DClass(L);
true
gap> DClass(H) = DClass(L);
true</pre></div>

<a id="X81B7AD4C7C552867" name="X81B7AD4C7C552867"></a>

<h5>10.1-2 GreensXClassOfElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensDClassOfElement</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DClass</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensHClassOfElement</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensHClassOfElement</code>( <var class="Arg">R</var>, <var class="Arg">i</var>, <var class="Arg">j</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HClass</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HClass</code>( <var class="Arg">R</var>, <var class="Arg">i</var>, <var class="Arg">j</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensLClassOfElement</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LClass</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensRClassOfElement</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RClass</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A Green's class.

These functions produce essentially the same output as the GAP library functions with the same names; see <code class="func">GreensDClassOfElement</code> (<a href="../../../doc/ref/chap51_mj.html#X87C75A9D86122D93">Reference: GreensDClassOfElement</a>). The main difference is that these functions can be applied to a wider class of objects:

<dl>
<dt><code class="code">GreensDClassOfElement</code> and <code class="code">DClass</code></dt>
<dd><var class="Arg">X</var> must be a semigroup.

</dd>
<dt><code class="code">GreensHClassOfElement</code> and <code class="code">HClass</code></dt>
<dd><var class="Arg">X</var> can be a semigroup, R-class, L-class, or D-class.

</dd>
<dd>If <var class="Arg">R</var> is a <var class="Arg">IxJ</var> Rees matrix semigroup or a Rees 0-matrix semigroup, and <var class="Arg">i</var> and <var class="Arg">j</var> are integers of the corresponding index sets, then <code class="code">GreensHClassOfElement</code> returns the H-class in row <var class="Arg">i</var> and column <var class="Arg">j</var>.

</dd>
<dt><code class="code">GreensLClassOfElement</code> and <code class="code">LClass</code></dt>
<dd><var class="Arg">X</var> can be a semigroup or D-class.

</dd>
<dt><code class="code">GreensRClassOfElement</code> and <code class="code">RClass</code></dt>
<dd><var class="Arg">X</var> can be a semigroup or D-class.

</dd>
</dl>
Note that <code class="code">GreensXClassOfElement</code> and <code class="code">XClass</code> are synonyms and have identical output. The shorter command is provided for the sake of convenience.

<a id="X7B44317786571F8B" name="X7B44317786571F8B"></a>

<h5>10.1-3 GreensXClassOfElementNC</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensDClassOfElementNC</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DClassNC</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensHClassOfElementNC</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HClassNC</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensLClassOfElementNC</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LClassNC</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensRClassOfElementNC</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RClassNC</code>( <var class="Arg">X</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A Green's class.

These functions are essentially the same as <code class="func">GreensDClassOfElement</code> (<a href="chap10.html#X81B7AD4C7C552867">10.1-2</a>) except that no effort is made to verify if <var class="Arg">f</var> is an element of <var class="Arg">X</var>. More precisely, <code class="code">GreensXClassOfElementNC</code> and <code class="code">XClassNC</code> first check if <var class="Arg">f</var> has already been shown to be an element of <var class="Arg">X</var>. If it is not known to GAP if <var class="Arg">f</var> is an element of <var class="Arg">X</var>, then no further attempt to verify this is made.

Note that <code class="code">GreensXClassOfElementNC</code> and <code class="code">XClassNC</code> are synonyms and have identical output. The shorter command is provided for the sake of convenience.

It can be quicker to compute the class of an element using <code class="code">GreensRClassOfElementNC</code>, say, than using <code class="code">GreensRClassOfElement</code> if it is known a priori that <var class="Arg">f</var> is an element of <var class="Arg">X</var>. On the other hand, if <var class="Arg">f</var> is not an element of <var class="Arg">X</var>, then the results of this computation are unpredictable.

For example, if

<div class="example"><pre>x := Transformation([15, 18, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20]);</pre></div>

in the semigroup <var class="Arg">X</var> of order-preserving mappings on 20 points, then

<div class="example"><pre>GreensRClassOfElementNC(X, x);</pre></div>

returns an answer relatively quickly, whereas

<div class="example"><pre>GreensRClassOfElement(X, x)</pre></div>

can take a significant amount of time to return a value.

See also <code class="func">GreensRClassOfElement</code> (<a href="../../../doc/ref/chap51_mj.html#X87C75A9D86122D93">Reference: GreensRClassOfElement</a>) and <code class="func">RClassOfHClass</code> (<a href="chap10.html#X87558FEF805D24E1">10.1-1</a>).

<div class="example"><pre>
gap> S := RandomSemigroup(IsTransformationSemigroup, 2, 1000);;
gap> x := [1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1];;
gap> x := EvaluateWord(Generators(S), x);;
gap> R := GreensRClassOfElementNC(S, x);;
gap> Size(R);
1
gap> L := GreensLClassOfElementNC(S, x);;
gap> Size(L);
1
gap> x := PartialPerm([2, 3, 4, 5, 0, 0, 6, 8, 10, 11]);;
gap> L := LClass(POI(11), x);
<Green's L-class: [1,2,3,4,5][7,6][9,10,11](8)>
gap> Size(L);
165</pre></div>

<a id="X7D51218A80234DE5" name="X7D51218A80234DE5"></a>

<h5>10.1-4 GreensXClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensDClasses</code>( <var class="Arg">obj</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DClasses</code>( <var class="Arg">obj</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensHClasses</code>( <var class="Arg">obj</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HClasses</code>( <var class="Arg">obj</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensJClasses</code>( <var class="Arg">obj</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ JClasses</code>( <var class="Arg">obj</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensLClasses</code>( <var class="Arg">obj</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LClasses</code>( <var class="Arg">obj</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensRClasses</code>( <var class="Arg">obj</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RClasses</code>( <var class="Arg">obj</var> )</td><td class="tdright">( method )</td></tr></table></div>
Returns: A list of Green's classes.

These functions produce essentially the same output as the GAP library functions with the same names; see <code class="func">GreensDClasses</code> (<a href="../../../doc/ref/chap51_mj.html#X844D20467A644811">Reference: GreensDClasses</a>). The main difference is that these functions can be applied to a wider class of objects:

<dl>
<dt><code class="code">GreensDClasses</code> and <code class="code">DClasses</code></dt>
<dd><var class="Arg">X</var> should be a semigroup.

</dd>
<dt><code class="code">GreensHClasses</code> and <code class="code">HClasses</code></dt>
<dd><var class="Arg">X</var> can be a semigroup, R-class, L-class, or D-class.

</dd>
<dt><code class="code">GreensLClasses</code> and <code class="code">LClasses</code></dt>
<dd><var class="Arg">X</var> can be a semigroup or D-class.

</dd>
<dt><code class="code">GreensRClasses</code> and <code class="code">RClasses</code></dt>
<dd><var class="Arg">X</var> can be a semigroup or D-class.

</dd>
</dl>
Note that <code class="code">GreensXClasses</code> and <code class="code">XClasses</code> are synonyms and have identical output. The shorter command is provided for the sake of convenience.

See also <code class="func">DClassReps</code> (<a href="chap10.html#X865387A87FAAC395">10.1-5</a>), <code class="func">IteratorOfDClassReps</code> (<a href="chap10.html#X8566F84A7F6D4193">10.2-1</a>), <code class="func">IteratorOfDClasses</code> (<a href="chap10.html#X867D7B8982915960">10.2-2</a>), and <code class="func">NrDClasses</code> (<a href="chap10.html#X7E45FD9F7BADDFBD">10.1-9</a>).

<div class="example"><pre>
gap> S := Semigroup(Transformation([3, 4, 4, 4]),
> Transformation([4, 3, 1, 2]));;
gap> GreensDClasses(S);
[ <Green's D-class: Transformation( [ 3, 4, 4, 4 ] )>,
 <Green's D-class: Transformation( [ 4, 3, 1, 2 ] )>,
 <Green's D-class: Transformation( [ 4, 4, 4, 4 ] )> ]
gap> GreensRClasses(S);
[ <Green's R-class: Transformation( [ 3, 4, 4, 4 ] )>,
 <Green's R-class: Transformation( [ 4, 3, 1, 2 ] )>,
 <Green's R-class: Transformation( [ 4, 4, 4, 4 ] )>,
 <Green's R-class: Transformation( [ 4, 4, 3, 4 ] )>,
 <Green's R-class: Transformation( [ 4, 3, 4, 4 ] )>,
 <Green's R-class: Transformation( [ 4, 4, 4, 3 ] )> ]
gap> D := GreensDClasses(S)[1];
<Green's D-class: Transformation( [ 3, 4, 4, 4 ] )>
gap> GreensLClasses(D);
[ <Green's L-class: Transformation( [ 3, 4, 4, 4 ] )>,
 <Green's L-class: Transformation( [ 1, 2, 2, 2 ] )> ]
gap> GreensRClasses(D);
[ <Green's R-class: Transformation( [ 3, 4, 4, 4 ] )>,
 <Green's R-class: Transformation( [ 4, 4, 3, 4 ] )>,
 <Green's R-class: Transformation( [ 4, 3, 4, 4 ] )>,
 <Green's R-class: Transformation( [ 4, 4, 4, 3 ] )> ]
gap> R := GreensRClasses(D)[1];
<Green's R-class: Transformation( [ 3, 4, 4, 4 ] )>
gap> GreensHClasses(R);
[ <Green's H-class: Transformation( [ 3, 4, 4, 4 ] )>,
 <Green's H-class: Transformation( [ 1, 2, 2, 2 ] )> ]
gap> S := InverseSemigroup([
> PartialPerm([2, 4, 1]), PartialPerm([3, 0, 4, 1])]);;
gap> GreensDClasses(S);
[ <Green's D-class: <identity partial perm on [ 1, 2, 4 ]>>,
 <Green's D-class: <identity partial perm on [ 1, 3, 4 ]>>,
 <Green's D-class: <identity partial perm on [ 1, 3 ]>>,
 <Green's D-class: <identity partial perm on [ 4 ]>>,
 <Green's D-class: <empty partial perm>> ]
gap> GreensLClasses(S);
[ <Green's L-class: <identity partial perm on [ 1, 2, 4 ]>>,
 <Green's L-class: [4,2,1,3]>,
 <Green's L-class: <identity partial perm on [ 1, 3, 4 ]>>,
 <Green's L-class: <identity partial perm on [ 1, 3 ]>>,
 <Green's L-class: [3,1,2]>, <Green's L-class: [1,4][3,2]>,
 <Green's L-class: [1,3,4]>, <Green's L-class: [3,1,4]>,
 <Green's L-class: [1,2](3)>,
 <Green's L-class: <identity partial perm on [ 4 ]>>,
 <Green's L-class: [4,1]>, <Green's L-class: [4,3]>,
 <Green's L-class: [4,2]>, <Green's L-class: <empty partial perm>> ]
gap> D := GreensDClasses(S)[3];
<Green's D-class: <identity partial perm on [ 1, 3 ]>>
gap> GreensLClasses(D);
[ <Green's L-class: <identity partial perm on [ 1, 3 ]>>,
 <Green's L-class: [3,1,2]>, <Green's L-class: [1,4][3,2]>,
 <Green's L-class: [1,3,4]>, <Green's L-class: [3,1,4]>,
 <Green's L-class: [1,2](3)> ]
gap> GreensRClasses(D);
[ <Green's R-class: <identity partial perm on [ 1, 3 ]>>,
 <Green's R-class: [2,1,3]>, <Green's R-class: [2,3][4,1]>,
 <Green's R-class: [4,3,1]>, <Green's R-class: [4,1,3]>,
 <Green's R-class: [2,1](3)> ]</pre></div>

<a id="X865387A87FAAC395" name="X865387A87FAAC395"></a>

<h5>10.1-5 XClassReps</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DClassReps</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HClassReps</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LClassReps</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RClassReps</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A list of representatives.

<code class="code">XClassReps</code> returns a list of the representatives of the Green's classes of <var class="Arg">obj</var>, which can be a semigroup, D-, L-, or R-class where appropriate.

The same output can be obtained by calling, for example:

<div class="example"><pre>List(GreensXClasses(obj), Representative);</pre></div>

Note that if the Green's classes themselves are not required, then <code class="code">XClassReps</code> will return an answer more quickly than the above, since the Green's class objects are not created.

See also <code class="func">GreensDClasses</code> (<a href="chap10.html#X7D51218A80234DE5">10.1-4</a>), <code class="func">IteratorOfDClassReps</code> (<a href="chap10.html#X8566F84A7F6D4193">10.2-1</a>), <code class="func">IteratorOfDClasses</code> (<a href="chap10.html#X867D7B8982915960">10.2-2</a>), and <code class="func">NrDClasses</code> (<a href="chap10.html#X7E45FD9F7BADDFBD">10.1-9</a>).

<div class="example"><pre>
gap> S := Semigroup(Transformation([3, 4, 4, 4]),
> Transformation([4, 3, 1, 2]));;
gap> DClassReps(S);
[ Transformation( [ 3, 4, 4, 4 ] ), Transformation( [ 4, 3, 1, 2 ] ),
 Transformation( [ 4, 4, 4, 4 ] ) ]
gap> LClassReps(S);
[ Transformation( [ 3, 4, 4, 4 ] ), Transformation( [ 1, 2, 2, 2 ] ),
 Transformation( [ 4, 3, 1, 2 ] ), Transformation( [ 4, 4, 4, 4 ] ),
 Transformation( [ 2, 2, 2, 2 ] ), Transformation( [ 3, 3, 3, 3 ] ),
 Transformation( [ 1, 1, 1, 1 ] ) ]
gap> D := GreensDClasses(S)[1];
<Green's D-class: Transformation( [ 3, 4, 4, 4 ] )>
gap> LClassReps(D);
[ Transformation( [ 3, 4, 4, 4 ] ), Transformation( [ 1, 2, 2, 2 ] ) ]
gap> RClassReps(D);
[ Transformation( [ 3, 4, 4, 4 ] ), Transformation( [ 4, 4, 3, 4 ] ),
 Transformation( [ 4, 3, 4, 4 ] ), Transformation( [ 4, 4, 4, 3 ] ) ]
gap> R := GreensRClasses(D)[1];;
gap> HClassReps(R);
[ Transformation( [ 3, 4, 4, 4 ] ), Transformation( [ 1, 2, 2, 2 ] ) ]
gap> S := SymmetricInverseSemigroup(6);;
gap> e := InverseSemigroup(Idempotents(S));;
gap> M := MunnSemigroup(e);;
gap> L := LClassNC(M, PartialPerm([51, 63], [51, 47]));;
gap> HClassReps(L);
[ <identity partial perm on [ 47, 51 ]>, [27,47](51), [50,47](51),
 [64,47](51), [63,47](51), [59,47](51) ]</pre></div>

<a id="X81E5A04F7DA3A1E1" name="X81E5A04F7DA3A1E1"></a>

<h5>10.1-6 MinimalDClass</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalDClass</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: The minimal D-class of a semigroup.

The minimal ideal of a semigroup is the least ideal with respect to containment. <code class="code">MinimalDClass</code> returns the D-class corresponding to the minimal ideal of the semigroup <var class="Arg">S</var>. Equivalently, <code class="code">MinimalDClass</code> returns the minimal D-class with respect to the partial order of D-classes.

It is significantly easier to find the minimal D-class of a semigroup, than to find its D-classes.

See also <code class="func">PartialOrderOfDClasses</code> (<a href="chap10.html#X8140814084748101">10.1-10</a>), <code class="func">IsGreensLessThanOrEqual</code> (<a href="../../../doc/ref/chap51_mj.html#X7AA204C8850F9070">Reference: IsGreensLessThanOrEqual</a>), <code class="func">MinimalIdeal</code> (<a href="chap11.html#X7BC68589879C3BE9">11.8-1</a>) and <code class="func">RepresentativeOfMinimalIdeal</code> (<a href="chap11.html#X7CA6744182D07C5B">11.8-2</a>).

<div class="example"><pre>
gap> D := MinimalDClass(JonesMonoid(8));
<Green's D-class: <bipartition: [ 1, 2 ], [ 3, 4 ], [ 5, 6 ],
 [ 7, 8 ], [ -1, -2 ], [ -3, -4 ], [ -5, -6 ], [ -7, -8 ]>>
gap> S := InverseSemigroup(
> PartialPerm([1, 2, 3, 5, 7, 8, 9], [2, 6, 9, 1, 5, 3, 8]),
> PartialPerm([1, 3, 4, 5, 7, 8, 9], [9, 4, 10, 5, 6, 7, 1]));;
gap> MinimalDClass(S);
<Green's D-class: <empty partial perm>></pre></div>

<a id="X834172F4787A565B" name="X834172F4787A565B"></a>

<h5>10.1-7 MaximalXClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalDClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalLClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalRClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: The maximal D, L, or R-classes of a semigroup.

Let <code class="code">X</code> be one of Green's D-, L-, or R-relations. Then <code class="code">MaximalXClasses</code> returns the maximal Green's <code class="code">X</code>-classes with respect to the partial order of <code class="code">X</code>-classes.

See also <code class="func">PartialOrderOfDClasses</code> (<a href="chap10.html#X8140814084748101">10.1-10</a>), <code class="func">IsGreensLessThanOrEqual</code> (<a href="../../../doc/ref/chap51_mj.html#X7AA204C8850F9070">Reference: IsGreensLessThanOrEqual</a>), and <code class="func">MinimalDClass</code> (<a href="chap10.html#X81E5A04F7DA3A1E1">10.1-6</a>).

<div class="example"><pre>
gap> MaximalDClasses(BrauerMonoid(8));
[ <Green's D-class: <block bijection: [ 1, -1 ], [ 2, -2 ],
 [ 3, -3 ], [ 4, -4 ], [ 5, -5 ], [ 6, -6 ], [ 7, -7 ],
 [ 8, -8 ]>> ]
gap> MaximalDClasses(FullTransformationMonoid(5));
[ <Green's D-class: IdentityTransformation> ]
gap> S := Semigroup(
> PartialPerm([1, 2, 3, 4, 5, 6, 7], [3, 8, 1, 4, 5, 6, 7]),
> PartialPerm([1, 2, 3, 6, 8], [2, 6, 7, 1, 5]),
> PartialPerm([1, 2, 3, 4, 6, 8], [4, 3, 2, 7, 6, 5]),
> PartialPerm([1, 2, 4, 5, 6, 7, 8], [7, 1, 4, 2, 5, 6, 3]));;
gap> MaximalDClasses(S);
[ <Green's D-class: [2,8](1,3)(4)(5)(6)(7)>,
 <Green's D-class: [8,3](1,7,6,5,2)(4)> ]</pre></div>

<a id="X7AA3F0A77D0043FB" name="X7AA3F0A77D0043FB"></a>

<h5>10.1-8 NrRegularDClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrRegularDClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RegularDClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A positive integer, or a list.

<code class="code">NrRegularDClasses</code> returns the number of regular D-classes of the semigroup <var class="Arg">S</var>.

<code class="code">RegularDClasses</code> returns a list of the regular D-classes of the semigroup <var class="Arg">S</var>.

See also <code class="func">IsRegularGreensClass</code> (<a href="chap10.html#X859DD1C079C80DCC">10.3-2</a>) and <code class="func">IsRegularDClass</code> (<a href="../../../doc/ref/chap51_mj.html#X7F5860927CAD920F">Reference: IsRegularDClass</a>).

<div class="example"><pre>
gap> S := Semigroup(Transformation([1, 3, 4, 1, 3, 5]),
> Transformation([5, 1, 6, 1, 6, 3]));;
gap> NrRegularDClasses(S);
3
gap> NrDClasses(S);
7
gap> AsSet(RegularDClasses(S));
[ <Green's D-class: Transformation( [ 1, 4, 1, 1, 4, 3 ] )>,
 <Green's D-class: Transformation( [ 1, 1, 1, 1, 1 ] )>,
 <Green's D-class: Transformation( [ 1, 1, 1, 1, 1, 1 ] )> ]</pre></div>

<a id="X7E45FD9F7BADDFBD" name="X7E45FD9F7BADDFBD"></a>

<h5>10.1-9 NrXClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrDClasses</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrHClasses</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrLClasses</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrRClasses</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A positive integer.

<code class="code">NrXClasses</code> returns the number of Green's classes in <var class="Arg">obj</var> where <var class="Arg">obj</var> can be a semigroup, D-, L-, or R-class where appropriate. If the actual Green's classes are not required, then it is more efficient to use

<div class="example"><pre>NrHClasses(obj)</pre></div>

than

<div class="example"><pre>Length(HClasses(obj))</pre></div>

since the Green's classes themselves are not created when <code class="code">NrXClasses</code> is called.

See also <code class="func">GreensRClasses</code> (<a href="chap10.html#X7D51218A80234DE5">10.1-4</a>), <code class="func">GreensRClasses</code> (<a href="../../../doc/ref/chap51_mj.html#X844D20467A644811">Reference: GreensRClasses</a>), <code class="func">IteratorOfRClasses</code> (<a href="chap10.html#X867D7B8982915960">10.2-2</a>), and

<div class="example"><pre>
gap> S := Semigroup(
> Transformation([1, 2, 5, 4, 3, 8, 7, 6]),
> Transformation([1, 6, 3, 4, 7, 2, 5, 8]),
> Transformation([2, 1, 6, 7, 8, 3, 4, 5]),
> Transformation([3, 2, 3, 6, 1, 6, 1, 2]),
> Transformation([5, 2, 3, 6, 3, 4, 7, 4]));;
gap> x := Transformation([2, 5, 4, 7, 4, 3, 6, 3]);;
gap> R := RClass(S, x);
<Green's R-class: Transformation( [ 2, 5, 4, 7, 4, 3, 6, 3 ] )>
gap> NrHClasses(R);
12
gap> D := DClass(R);
<Green's D-class: Transformation( [ 2, 5, 4, 7, 4, 3, 6, 3 ] )>
gap> NrHClasses(D);
72
gap> L := LClass(S, x);
<Green's L-class: Transformation( [ 2, 5, 4, 7, 4, 3, 6, 3 ] )>
gap> NrHClasses(L);
6
gap> NrHClasses(S);
1555
gap> S := Semigroup(Transformation([4, 6, 5, 2, 1, 3]),
> Transformation([6, 3, 2, 5, 4, 1]),
> Transformation([1, 2, 4, 3, 5, 6]),
> Transformation([3, 5, 6, 1, 2, 3]),
> Transformation([5, 3, 6, 6, 6, 2]),
> Transformation([2, 3, 2, 6, 4, 6]),
> Transformation([2, 1, 2, 2, 2, 4]),
> Transformation([4, 4, 1, 2, 1, 2]));;
gap> NrRClasses(S);
150
gap> Size(S);
6342
gap> x := Transformation([1, 3, 3, 1, 3, 5]);;
gap> D := DClass(S, x);
<Green's D-class: Transformation( [ 1, 3, 3, 1, 3, 5 ] )>
gap> NrRClasses(D);
87
gap> S := SymmetricInverseSemigroup(10);;
gap> NrDClasses(S); NrRClasses(S); NrHClasses(S); NrLClasses(S);
11
1024
184756
1024
gap> S := POPI(10);;
gap> NrDClasses(S);
11
gap> NrRClasses(S);
1024</pre></div>

<a id="X8140814084748101" name="X8140814084748101"></a>

<h5>10.1-10 PartialOrderOfXClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PartialOrderOfDClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PartialOrderOfLClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PartialOrderOfRClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A digraph.

Let <code class="code">X</code> be one of Green's D-, L-, or R-relations. Then <code class="code">PartialOrderOfXClasses</code> returns a digraph <code class="code">D</code> where <code class="code">OutNeighbours(D)[i]</code> contains every <code class="code">j</code> such that <code class="code">GreensXClasses(S)[j]</code> is immediately less than <code class="code">GreensXClasses(S)[i]</code> in the partial order of <code class="code">X</code>-classes of <var class="Arg">S</var>. The reflexive transitive closure of the digraph <code class="code">D</code> is the partial order of <code class="code">X</code>-classes of <var class="Arg">S</var> (in the sense of the digraphs package).

The partial order on the <code class="code">X</code>-classes is defined as follows.

<dl>
<dt>Green's D-relation:</dt>
<dd>x≤ y if and only if S ^ 1xS ^ 1 is a subset of S ^ 1yS ^ 1.

</dd>
<dt>Green's L-relation:</dt>
<dd>x≤ y if and only if S ^ 1x is a subset of S ^ 1y.

</dd>
<dt>Green's R-relation:</dt>
<dd>x≤ y if and only if xS ^ 1 is a subset of yS ^ 1.

</dd>
</dl>
See also <code class="func">GreensDClasses</code> (<a href="chap10.html#X7D51218A80234DE5">10.1-4</a>), <code class="func">GreensDClasses</code> (<a href="../../../doc/ref/chap51_mj.html#X844D20467A644811">Reference: GreensDClasses</a>), <code class="func">IsGreensLessThanOrEqual</code> (<a href="../../../doc/ref/chap51_mj.html#X7AA204C8850F9070">Reference: IsGreensLessThanOrEqual</a>), and <code class="func">\<</code> (<a href="chap10.html#X85F30ACF86C3A733">10.3-1</a>).

<div class="example"><pre>
gap> S := Semigroup(Transformation([2, 4, 1, 2]),
> Transformation([3, 3, 4, 1]));;
gap> PartialOrderOfDClasses(S);
<immutable digraph with 4 vertices, 3 edges>
gap> IsGreensLessThanOrEqual(GreensDClasses(S)[1],
> GreensDClasses(S)[2]);
false
gap> IsGreensLessThanOrEqual(GreensDClasses(S)[2],
> GreensDClasses(S)[1]);
false
gap> IsGreensLessThanOrEqual(GreensDClasses(S)[3],
> GreensDClasses(S)[1]);
true
gap> S := InverseSemigroup(
> PartialPerm([1, 2, 3], [1, 3, 4]),
> PartialPerm([1, 3, 5], [5, 1, 3]));;
gap> Size(S);
58
gap> PartialOrderOfDClasses(S);
<immutable digraph with 5 vertices, 4 edges>
gap> IsGreensLessThanOrEqual(GreensDClasses(S)[1],
> GreensDClasses(S)[2]);
false
gap> IsGreensLessThanOrEqual(GreensDClasses(S)[5],
> GreensDClasses(S)[2]);
true
gap> IsGreensLessThanOrEqual(GreensDClasses(S)[3],
> GreensDClasses(S)[4]);
false
gap> IsGreensLessThanOrEqual(GreensDClasses(S)[4],
> GreensDClasses(S)[3]);
true</pre></div>

<a id="X83B0EDA57F1D2F97" name="X83B0EDA57F1D2F97"></a>

<h5>10.1-11 LengthOfLongestDClassChain</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LengthOfLongestDClassChain</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A non-negative integer.

If <var class="Arg">S</var> is a semigroup, then <code class="code">LengthOfLongestDClassChain</code> returns the length of the longest chain in the partial order defined by <code class="code">PartialOrderOfDClasses(<var class="Arg">S</var>)</code>. See <code class="func">PartialOrderOfDClasses</code> (<a href="chap10.html#X8140814084748101">10.1-10</a>).

The partial order on the D-classes is defined by x≤ y if and only if S ^ 1xS ^ 1 is a subset of S ^ 1yS ^ 1. A chain of D-classes is a collection of <code class="code">n</code> D-classes D_1, D_2, ... D_n such that D_1 < D_2 < ⋯ < D_n. The length of such a chain is <code class="code">n - 1</code>.

<div class="example"><pre>
gap> S := TrivialSemigroup();;
gap> LengthOfLongestDClassChain(S);
0
gap> T := ZeroSemigroup(5);;
gap> LengthOfLongestDClassChain(T);
1
gap> U := MonogenicSemigroup(14, 7);;
gap> LengthOfLongestDClassChain(U);
13
gap> V := FullTransformationMonoid(6);
<full transformation monoid of degree 6>
gap> LengthOfLongestDClassChain(V);
5</pre></div>

<a id="X7E872C5381D0DD8A" name="X7E872C5381D0DD8A"></a>

<h5>10.1-12 IsGreensDGreaterThanFunc</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensDGreaterThanFunc</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A function.

<code class="code">IsGreensDGreaterThanFunc(<var class="Arg">S</var>)</code> returns a function <code class="code">func</code> such that for any two elements <code class="code">x</code> and <code class="code">y</code> of <var class="Arg">S</var>, <code class="code">func(x, y)</code> return <code class="keyw">true</code> if the D-class of <code class="code">x</code> in <var class="Arg">S</var> is greater than or equal to the D-class of <code class="code">y</code> in <var class="Arg">S</var> under the usual ordering of Green's D-classes of a semigroup.

<div class="example"><pre>
gap> S := Semigroup(Transformation([1, 3, 4, 1, 3]),
> Transformation([2, 4, 1, 5, 5]),
> Transformation([2, 5, 3, 5, 3]),
> Transformation([5, 5, 1, 1, 3]));;
gap> reps := ShallowCopy(AsSet(DClassReps(S)));
[ Transformation( [ 1, 1, 1, 1, 1 ] ),
 Transformation( [ 1, 3, 1, 3, 3 ] ),
 Transformation( [ 1, 3, 4, 1, 3 ] ),
 Transformation( [ 2, 4, 1, 5, 5 ] ) ]
gap> Sort(reps, IsGreensDGreaterThanFunc(S));
gap> reps;
[ Transformation( [ 2, 4, 1, 5, 5 ] ),
 Transformation( [ 1, 3, 4, 1, 3 ] ),
 Transformation( [ 1, 3, 1, 3, 3 ] ),
 Transformation( [ 1, 1, 1, 1, 1 ] ) ]
gap> IsGreensLessThanOrEqual(DClass(S, reps[2]),
> DClass(S, reps[1]));
true
gap> S := DualSymmetricInverseMonoid(4);;
gap> IsGreensDGreaterThanFunc(S)(S.1, S.3);
true
gap> IsGreensDGreaterThanFunc(S)(S.3, S.1);
false
gap> IsGreensLessThanOrEqual(DClass(S, S.3),
> DClass(S, S.1));
true
gap> IsGreensLessThanOrEqual(DClass(S, S.1),
> DClass(S, S.3));
false</pre></div>

<a id="X819CCBD67FD27115" name="X819CCBD67FD27115"></a>

<h4>10.2 
 Iterators and enumerators of classes and representatives
 </h4>

In this section, we describe the methods in the Semigroups package for incrementally determining Green's classes or their representatives.

<a id="X8566F84A7F6D4193" name="X8566F84A7F6D4193"></a>

<h5>10.2-1 IteratorOfXClassReps</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IteratorOfDClassReps</code>( <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IteratorOfHClassReps</code>( <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IteratorOfLClassReps</code>( <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: An iterator.

Returns an iterator of the representatives of the Green's classes contained in the semigroup <var class="Arg">S</var>. See <a href="../../../doc/ref/chap30_mj.html#X85A3F00985453F95">Reference: Iterators</a> for more information on iterators.

See also <code class="func">GreensRClasses</code> (<a href="../../../doc/ref/chap51_mj.html#X844D20467A644811">Reference: GreensRClasses</a>), <code class="func">GreensRClasses</code> (<a href="chap10.html#X7D51218A80234DE5">10.1-4</a>), and <code class="func">IteratorOfRClasses</code> (<a href="chap10.html#X867D7B8982915960">10.2-2</a>).

<a id="X867D7B8982915960" name="X867D7B8982915960"></a>

<h5>10.2-2 IteratorOfXClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IteratorOfDClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IteratorOfRClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: An iterator.

Returns an iterator of the Green's classes in the semigroup <var class="Arg">S</var>. See <a href="../../../doc/ref/chap30_mj.html#X85A3F00985453F95">Reference: Iterators</a> for more information on iterators.

This function is useful if you are, for example, looking for an R-class of a semigroup with a particular property but do not necessarily want to compute all of the R-classes.

See also <code class="func">GreensRClasses</code> (<a href="chap10.html#X7D51218A80234DE5">10.1-4</a>), <code class="func">GreensRClasses</code> (<a href="../../../doc/ref/chap51_mj.html#X844D20467A644811">Reference: GreensRClasses</a>), and <code class="func">NrRClasses</code> (<a href="chap10.html#X7E45FD9F7BADDFBD">10.1-9</a>).

The transformation semigroup in the example below has 25147892 elements but it only takes a fraction of a second to find a non-trivial R-class. The inverse semigroup of partial permutations in the example below has size 158122047816 but it only takes a fraction of a second to find an R-class with more than 1000 elements.

<div class="example"><pre>
gap> gens := [Transformation([2, 4, 1, 5, 4, 4, 7, 3, 8, 1]),
> Transformation([3, 2, 8, 8, 4, 4, 8, 6, 5, 7]),
> Transformation([4, 10, 6, 6, 1, 2, 4, 10, 9, 7]),
> Transformation([6, 2, 2, 4, 9, 9, 5, 10, 1, 8]),
> Transformation([6, 4, 1, 6, 6, 8, 9, 6, 2, 2]),
> Transformation([6, 8, 1, 10, 6, 4, 9, 1, 9, 4]),
> Transformation([8, 6, 2, 3, 3, 4, 8, 6, 2, 9]),
> Transformation([9, 1, 2, 8, 1, 5, 9, 9, 9, 5]),
> Transformation([9, 3, 1, 5, 10, 3, 4, 6, 10, 2]),
> Transformation([10, 7, 3, 7, 1, 9, 8, 8, 4, 10])];;
gap> S := Semigroup(gens);;
gap> iter := IteratorOfRClasses(S);
<iterator>
gap> for R in iter do
> if Size(R) > 1 then
> break;
> fi;
> od;
gap> R;
<Green's R-class: Transformation( [ 6, 4, 1, 6, 6, 8, 9, 6, 2, 2 ] )>
gap> Size(R);
21600
gap> S := InverseSemigroup(
> PartialPerm([1, 2, 3, 4, 5, 6, 7, 10, 11, 19, 20],
> [19, 4, 11, 15, 3, 20, 1, 14, 8, 13, 17]),
> PartialPerm([1, 2, 3, 4, 6, 7, 8, 14, 15, 16, 17],
> [15, 14, 20, 19, 4, 5, 1, 13, 11, 10, 3]),
> PartialPerm([1, 2, 4, 6, 7, 8, 9, 10, 14, 15, 18],
> [7, 2, 17, 10, 1, 19, 9, 3, 11, 16, 18]),
> PartialPerm([1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16],
> [8, 3, 18, 1, 4, 13, 12, 7, 19, 20, 2, 11]),
> PartialPerm([1, 2, 3, 4, 5, 6, 7, 9, 11, 15, 16, 17, 20],
> [7, 17, 13, 4, 6, 9, 18, 10, 11, 19, 5, 2, 8]),
> PartialPerm([1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 18],
> [10, 20, 11, 7, 13, 8, 4, 9, 2, 18, 17, 6, 15]),
> PartialPerm([1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 17, 18],
> [10, 20, 18, 1, 14, 16, 9, 5, 15, 4, 8, 12, 19, 11]),
> PartialPerm([1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 15, 16, 19, 20],
> [13, 6, 1, 2, 11, 7, 16, 18, 9, 10, 4, 14, 15, 5, 17]),
> PartialPerm([1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 20],
> [5, 3, 12, 9, 20, 15, 8, 16, 13, 1, 17, 11, 14, 10, 2]),
> PartialPerm([1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 13, 17, 18, 19, 20],
> [8, 3, 9, 20, 2, 12, 14, 15, 4, 18, 13, 1, 17, 19, 5]));;
gap> iter := IteratorOfRClasses(S);
<iterator>
gap> repeat
> R := NextIterator(iter);
> until Size(R) > 1000;
gap> R;
<Green's R-class: [8,19,14][11,4][13,15,5][17,20]>
gap> Size(R);
10020240</pre></div>

<a id="X820EF2BA7D5D53B4" name="X820EF2BA7D5D53B4"></a>

<h4>10.3 
 Properties of Green's classes
 </h4>

In this section, we describe the properties and operators of Green's classes that are available in the Semigroups package.

<a id="X85F30ACF86C3A733" name="X85F30ACF86C3A733"></a>

<h5>10.3-1 Less than for Green's classes</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \<</code>( <var class="Arg">left-expr</var>, <var class="Arg">right-expr</var> )</td><td class="tdright">( method )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

The Green's class <var class="Arg">left-expr</var> is less than or equal to <var class="Arg">right-expr</var> if they belong to the same semigroup and the representative of <var class="Arg">left-expr</var> is less than the representative of <var class="Arg">right-expr</var> under <code class="code"><</code>; see also <code class="func">Representative</code> (<a href="../../../doc/ref/chap30_mj.html#X865507568182424E">Reference: Representative</a>).

Please note that this is not the usual order on the Green's classes of a semigroup as defined in <a href="../../../doc/ref/chap51_mj.html#X80C6C718801855E9">Reference: Green's Relations</a>. See also <code class="func">IsGreensLessThanOrEqual</code> (<a href="../../../doc/ref/chap51_mj.html#X7AA204C8850F9070">Reference: IsGreensLessThanOrEqual</a>).

<div class="example"><pre>
gap> S := FullTransformationSemigroup(4);;
gap> A := GreensRClassOfElement(S, Transformation([2, 1, 3, 1]));
<Green's R-class: Transformation( [ 2, 1, 3, 1 ] )>
gap> B := GreensRClassOfElement(S, Transformation([1, 2, 3, 4]));
<Green's R-class: IdentityTransformation>
gap> A < B;
false
gap> B < A;
true
gap> IsGreensLessThanOrEqual(A, B);
true
gap> IsGreensLessThanOrEqual(B, A);
false
gap> S := SymmetricInverseSemigroup(4);;
gap> A := GreensJClassOfElement(S, PartialPerm([1, 3, 4]));;
gap> B := GreensJClassOfElement(S, PartialPerm([3, 1]));;
gap> A < B;
true
gap> B < A;
false
gap> IsGreensLessThanOrEqual(A, B);
false
gap> IsGreensLessThanOrEqual(B, A);
true</pre></div>

<a id="X859DD1C079C80DCC" name="X859DD1C079C80DCC"></a>

<h5>10.3-2 IsRegularGreensClass</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRegularGreensClass</code>( <var class="Arg">class</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

This function returns <code class="keyw">true</code> if <var class="Arg">class</var> is a regular Green's class and <code class="keyw">false</code> if it is not. See also <code class="func">IsRegularDClass</code> (<a href="../../../doc/ref/chap51_mj.html#X7F5860927CAD920F">Reference: IsRegularDClass</a>), <code class="func">IsGroupHClass</code> (<a href="../../../doc/ref/chap51_mj.html#X79D740EF7F0E53BD">Reference: IsGroupHClass</a>), <code class="func">GroupHClassOfGreensDClass</code> (<a href="../../../doc/ref/chap51_mj.html#X7CB4A18685B850E2">Reference: GroupHClassOfGreensDClass</a>), <code class="func">GroupHClass</code> (<a href="chap10.html#X8723756387DD4C0F">10.4-1</a>), <code class="func">NrIdempotents</code> (<a href="chap11.html#X7CFC4DB387452320">11.10-2</a>), <code class="func">Idempotents</code> (<a href="chap11.html#X7C651C9C78398FFF">11.10-1</a>), and <code class="func">IsRegularSemigroupElement</code> (<a href="../../../doc/ref/chap51_mj.html#X87532A76854347E0">Reference: IsRegularSemigroupElement</a>).

The function <code class="code">IsRegularDClass</code> produces the same output as the GAP library functions with the same name; see <code class="func">IsRegularDClass</code> (<a href="../../../doc/ref/chap51_mj.html#X7F5860927CAD920F">Reference: IsRegularDClass</a>).

<div class="example"><pre>
gap> S := Monoid(Transformation([10, 8, 7, 4, 1, 4, 10, 10, 7, 2]),
> Transformation([5, 2, 5, 5, 9, 10, 8, 3, 8, 10]));;
gap> f := Transformation([1, 1, 10, 8, 8, 8, 1, 1, 10, 8]);;
gap> R := RClass(S, f);;
gap> IsRegularGreensClass(R);
true
gap> S := Monoid(Transformation([2, 3, 4, 5, 1, 8, 7, 6, 2, 7]),
> Transformation([3, 8, 7, 4, 1, 4, 3, 3, 7, 2]));;
gap> f := Transformation([3, 8, 7, 4, 1, 4, 3, 3, 7, 2]);;
gap> R := RClass(S, f);;
gap> IsRegularGreensClass(R);
false
gap> NrIdempotents(R);
0
gap> S := Semigroup(Transformation([2, 1, 3, 1]),
> Transformation([3, 1, 2, 1]),
> Transformation([4, 2, 3, 3]));;
gap> f := Transformation([4, 2, 3, 3]);;
gap> L := GreensLClassOfElement(S, f);;
gap> IsRegularGreensClass(L);
false
gap> R := GreensRClassOfElement(S, f);;
gap> IsRegularGreensClass(R);
false
gap> g := Transformation([4, 4, 4, 4]);;
gap> IsRegularSemigroupElement(S, g);
true
gap> IsRegularGreensClass(LClass(S, g));
true
gap> IsRegularGreensClass(RClass(S, g));
true
gap> IsRegularDClass(DClass(S, g));
true
gap> DClass(S, g) = RClass(S, g);
false</pre></div>

<a id="X7E9BD34B8021045A" name="X7E9BD34B8021045A"></a>

<h5>10.3-3 IsGreensClassNC</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensClassNC</code>( <var class="Arg">class</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

A Green's class <var class="Arg">class</var> of a semigroup <code class="code">S</code> satisfies <code class="code">IsGreensClassNC</code> if it was not known to GAP that the representative of <var class="Arg">class</var> was an element of <code class="code">S</code> at the point that <var class="Arg">class</var> was created.

<a id="X855723B17D4AAF8F" name="X855723B17D4AAF8F"></a>

<h4>10.4 
 Attributes of Green's classes
 </h4>

In this section, we describe the attributes of Green's classes that are available in the Semigroups package.

<a id="X8723756387DD4C0F" name="X8723756387DD4C0F"></a>

<h5>10.4-1 GroupHClass</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupHClass</code>( <var class="Arg">class</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A group H-class of the D-class <var class="Arg">class</var> if it is regular and <code class="keyw">fail</code> if it is not.

<code class="code">GroupHClass</code> is a synonym for <code class="func">GroupHClassOfGreensDClass</code> (<a href="../../../doc/ref/chap51_mj.html#X7CB4A18685B850E2">Reference: GroupHClassOfGreensDClass</a>).

See also <code class="func">IsGroupHClass</code> (<a href="../../../doc/ref/chap51_mj.html#X79D740EF7F0E53BD">Reference: IsGroupHClass</a>), <code class="func">IsRegularDClass</code> (<a href="../../../doc/ref/chap51_mj.html#X7F5860927CAD920F">Reference: IsRegularDClass</a>), <code class="func">IsRegularGreensClass</code> (<a href="chap10.html#X859DD1C079C80DCC">10.3-2</a>), and <code class="func">IsRegularSemigroup</code> (<a href="chap12.html#X7C4663827C5ACEF1">12.1-17</a>).

<div class="example"><pre>
gap> S := Semigroup(Transformation([2, 6, 7, 2, 6, 1, 1, 5]),
> Transformation([3, 8, 1, 4, 5, 6, 7, 1]));;
gap> IsRegularSemigroup(S);
false
gap> iter := IteratorOfDClasses(S);;
gap> repeat D := NextIterator(iter); until IsRegularDClass(D);
gap> D;
<Green's D-class: Transformation( [ 6, 1, 1, 6, 1, 2, 2, 6 ] )>
gap> NrIdempotents(D);
12
gap> NrRClasses(D);
8
gap> NrLClasses(D);
4
gap> GroupHClass(D);
<Green's H-class: Transformation( [ 1, 2, 2, 1, 2, 6, 6, 1 ] )>
gap> GroupHClassOfGreensDClass(D);
<Green's H-class: Transformation( [ 1, 2, 2, 1, 2, 6, 6, 1 ] )>
gap> StructureDescription(GroupHClass(D));
"S3"
gap> repeat D := NextIterator(iter); until not IsRegularDClass(D);
gap> D;
<Green's D-class: Transformation( [ 7, 5, 2, 2, 6, 1, 1, 2 ] )>
gap> IsRegularDClass(D);
false
gap> GroupHClass(D);
fail
gap> S := InverseSemigroup(
> PartialPerm([2, 1, 6, 0, 3]), PartialPerm([3, 5, 2, 0, 0, 6]));;
gap> x := PartialPerm([1 .. 3], [6, 3, 1]);;
gap> First(DClasses(S), x -> not IsTrivial(GroupHClass(x)));
<Green's D-class: <identity partial perm on [ 1, 2 ]>>
gap> StructureDescription(GroupHClass(last));
"C2"</pre></div>

<a id="X84F1321E8217D2A8" name="X84F1321E8217D2A8"></a>

<h5>10.4-2 SchutzenbergerGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SchutzenbergerGroup</code>( <var class="Arg">class</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A group.

<code class="code">SchutzenbergerGroup</code> returns the generalized Schutzenberger group (defined below) of the R-, D-, L-, or H-class <var class="Arg">class</var>.

If <code class="code">f</code> is an element of a semigroup of transformations or partial permutations and <code class="code">im(f)</code> denotes the image of <code class="code">f</code>, then the generalized Schutzenberger group of <code class="code">im(f)</code> is the permutation group


 \{\:g|_{\textrm{im}(f)}\::\:\textrm{im}(f*g)=\textrm{im}(f)\:\}.
 

The generalized Schutzenberger group of the kernel <code class="code">ker(f)</code> of a transformation <code class="code">f</code> or the domain <code class="code">dom(f)</code> of a partial permutation <code class="code">f</code> is defined analogously.

The generalized Schutzenberger group of a Green's class is then defined as follows.

<dl>
<dt>R-class</dt>
<dd>The generalized Schutzenberger group of the image or range of the representative of the R-class.

</dd>
<dt>L-class</dt>
<dd>The generalized Schutzenberger group of the kernel or domain of the representative of the L-class.

</dd>
<dt>H-class</dt>
<dd>The intersection of the generalized Schutzenberger groups of the R- and L-class containing the H-class.

</dd>
<dt>D-class</dt>
<dd>The intersection of the generalized Schutzenberger groups of the R- and L-class containing the representative of the D-class.

</dd>
</dl>
The output of this attribute is difficult to describe for other types of semigroup. However, a general description is given in <a href="chapBib.html#biBMitchell2019aa">[EEMP19]</a>.

<div class="example"><pre>
gap> S := Semigroup(Transformation([4, 4, 3, 5, 3]),
> Transformation([5, 1, 1, 4, 1]),
> Transformation([5, 5, 4, 4, 5]));;
gap> f := Transformation([5, 5, 4, 4, 5]);;
gap> SchutzenbergerGroup(RClass(S, f));
Group([ (4,5) ])
gap> S := InverseSemigroup(
> PartialPerm([1, 2, 3, 7],
> [9, 2, 4, 8]),
> PartialPerm([1, 2, 6, 7, 8, 9, 10],
> [6, 8, 4, 5, 9, 1, 3]),
> PartialPerm([1, 2, 3, 5, 6, 7, 8, 9],
> [7, 4, 1, 6, 9, 5, 2, 3]));;
gap> List(DClasses(S), SchutzenbergerGroup);
[ Group(()), Group(()), Group(()), Group(()), Group([ (4,9) ]),
 Group(()), Group(()), Group([ (5,8,6), (5,8) ]), Group(()),
 Group(()), Group(()), Group(()), Group(()), Group(()),
 Group([ (1,7,5,6,9,3) ]), Group([ (1,6)(3,5) ]), Group(()),
 Group(()), Group(()), Group(()), Group(()), Group(()), Group(()) ]</pre></div>

<a id="X81202126806443F9" name="X81202126806443F9"></a>

<h5>10.4-3 StructureDescriptionSchutzenbergerGroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StructureDescriptionSchutzenbergerGroups</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: Distinct structure descriptions of the Schutzenberger groups of a semigroup.

<code class="code">StructureDescriptionSchutzenbergerGroups</code> returns the distinct values of <code class="func">StructureDescription</code> (<a href="../../../doc/ref/chap39_mj.html#X8199B74B84446971">Reference: StructureDescription</a>) when it is applied to the Schutzenberger groups of the R-classes of the semigroup <var class="Arg">S</var>.

<div class="example"><pre>
gap> S := Semigroup([
> PartialPerm([1, 2, 3], [2, 5, 4]),
> PartialPerm([1, 2, 3], [4, 1, 2]),
> PartialPerm([1, 2, 3], [5, 2, 3]),
> PartialPerm([1, 2, 4, 5], [2, 1, 4, 3]),
> PartialPerm([1, 2, 5], [2, 3, 5]),
> PartialPerm([1, 2, 3, 5], [2, 3, 5, 4]),
> PartialPerm([1, 2, 3, 5], [4, 2, 5, 1]),
> PartialPerm([1, 2, 3, 5], [5, 2, 4, 3]),
> PartialPerm([1, 2, 5], [5, 4, 3])]);;
gap> StructureDescriptionSchutzenbergerGroups(S);
[ "1", "C2", "S3" ]
gap> S := Monoid(
> Bipartition([[1, 2, 5, -1, -2], [3, 4, -3, -5], [-4]]),
> Bipartition([[1, 2, -2], [3, -1], [4], [5], [-3, -4], [-5]]),
> Bipartition([[1], [2, 3, -5], [4, -3], [5, -2], [-1, -4]]));
<bipartition monoid of degree 5 with 3 generators>
gap> StructureDescriptionSchutzenbergerGroups(S);
[ "1", "C2" ]</pre></div>

<a id="X838F43FE79A8C678" name="X838F43FE79A8C678"></a>

<h5>10.4-4 StructureDescriptionMaximalSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StructureDescriptionMaximalSubgroups</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: Distinct structure descriptions of the maximal subgroups of a semigroup.

<code class="code">StructureDescriptionMaximalSubgroups</code> returns the distinct values of <code class="func">StructureDescription</code> (<a href="../../../doc/ref/chap39_mj.html#X8199B74B84446971">Reference: StructureDescription</a>) when it is applied to the maximal subgroups of the semigroup <var class="Arg">S</var>.

<div class="example"><pre>
gap> S := DualSymmetricInverseSemigroup(6);
<inverse block bijection monoid of degree 6 with 3 generators>
gap> StructureDescriptionMaximalSubgroups(S);
[ "1", "C2", "S3", "S4", "S5", "S6" ]
gap> S := Semigroup(
> PartialPerm([1, 3, 4, 5, 8],
> [8, 3, 9, 4, 5]),
> PartialPerm([1, 2, 3, 4, 8],
> [10, 4, 1, 9, 6]),
> PartialPerm([1, 2, 3, 4, 5, 6, 7, 10],
> [4, 1, 6, 7, 5, 3, 2, 10]),
> PartialPerm([1, 2, 3, 4, 6, 8, 10],
> [4, 9, 10, 3, 1, 5, 2]));;
gap> StructureDescriptionMaximalSubgroups(S);
[ "1", "C2", "C3", "C4" ]</pre></div>

<a id="X8459E4067C5773AD" name="X8459E4067C5773AD"></a>

<h5>10.4-5 MultiplicativeNeutralElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MultiplicativeNeutralElement</code>( <var class="Arg">H</var> )</td><td class="tdright">( method )</td></tr></table></div>
Returns: A semigroup element or <code class="keyw">fail</code>.

If the H-class <var class="Arg">H</var> of a semigroup <code class="code">S</code> is a subgroup of <code class="code">S</code>, then <code class="code">MultiplicativeNeutralElement</code> returns the identity of <var class="Arg">H</var>. If <var class="Arg">H</var> is not a subgroup of <code class="code">S</code>, then <code class="keyw">fail</code> is returned.

<div class="example"><pre>
gap> S := Semigroup([PartialPerm([1, 5, 2]),
> PartialPerm([2, 0, 4]), PartialPerm([4, 1, 5]),
> PartialPerm([1, 0, 3, 0, 4]), PartialPerm([1, 2, 0, 3, 5]),
> PartialPerm([1, 3, 2, 0, 5]), PartialPerm([5, 0, 0, 4, 3])]);;
gap> H := HClass(S, PartialPerm([1, 2]));;
gap> MultiplicativeNeutralElement(H);
<identity partial perm on [ 1, 2 ]>
gap> H := HClass(S, PartialPerm([1, 4]));;
gap> MultiplicativeNeutralElement(H);
fail</pre></div>

<a id="X85B34FFB82C83127" name="X85B34FFB82C83127"></a>

<h5>10.4-6 StructureDescription</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StructureDescription</code>( <var class="Arg">class</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A string or <code class="keyw">fail</code>.

<code class="code">StructureDescription</code> returns the value of <code class="func">StructureDescription</code> (<a href="../../../doc/ref/chap39_mj.html#X8199B74B84446971">Reference: StructureDescription</a>) when it is applied to a group isomorphic to the group H-class <var class="Arg">class</var>. If <var class="Arg">class</var> is not a group H-class, then <code class="keyw">fail</code> is returned.

<div class="example"><pre>
gap> S := Semigroup(
> PartialPerm([1, 2, 3, 4, 6, 7, 8, 9],
> [1, 9, 4, 3, 5, 2, 10, 7]),
> PartialPerm([1, 2, 4, 7, 8, 9],
> [6, 2, 4, 9, 1, 3]));;
gap> H := HClass(S, PartialPerm([1, 2, 3, 4, 7, 9],
> [1, 7, 3, 4, 9, 2]));;
gap> StructureDescription(H);
"C6"</pre></div>

<a id="X7EBB4F1981CC2AE9" name="X7EBB4F1981CC2AE9"></a>

<h5>10.4-7 InjectionPrincipalFactor</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InjectionPrincipalFactor</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InjectionNormalizedPrincipalFactor</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismReesMatrixSemigroup</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A injective mapping.

If the D-class <var class="Arg">D</var> is a subsemigroup of a semigroup <code class="code">S</code>, then the principal factor of <var class="Arg">D</var> is just <var class="Arg">D</var> itself. If <var class="Arg">D</var> is not a subsemigroup of <code class="code">S</code>, then the principal factor of <var class="Arg">D</var> is the semigroup with elements <var class="Arg">D</var> and a new element <code class="code">0</code> with multiplication of x,y∈ D defined by:


 xy=\left\{\begin{array}{ll}
 x*y\ (\textrm{in }S)&\textrm{if }x*y\in D\\
 0&\textrm{if }xy\not\in D.
 \end{array}\right.
 

<code class="code">InjectionPrincipalFactor</code> returns an injective function from the D-class <var class="Arg">D</var> to a Rees (0-)matrix semigroup, which contains the principal factor of <var class="Arg">D</var> as a subsemigroup.

If <var class="Arg">D</var> is a subsemigroup of its parent semigroup, then the function returned by <code class="code">InjectionPrincipalFactor</code> or <code class="code">IsomorphismReesMatrixSemigroup</code> is an isomorphism from <var class="Arg">D</var> to a Rees matrix semigroup; see <code class="func">ReesMatrixSemigroup</code> (<a href="../../../doc/ref/chap51_mj.html#X8526AA557CDF6C49">Reference: ReesMatrixSemigroup</a>).

If <var class="Arg">D</var> is not a semigroup, then the function returned by <code class="code">InjectionPrincipalFactor</code> is an injective function from <var class="Arg">D</var> to a Rees 0-matrix semigroup isomorphic to the principal factor of <var class="Arg">D</var>; see <code class="func">ReesZeroMatrixSemigroup</code> (<a href="../../../doc/ref/chap51_mj.html#X8526AA557CDF6C49">Reference: ReesZeroMatrixSemigroup</a>). In this case, <code class="code">IsomorphismReesMatrixSemigroup</code> and <code class="code">IsomorphismReesZeroMatrixSemigroup</code> returns an error.

<code class="code">InjectionNormalizedPrincipalFactor</code> returns the composition of <code class="code">InjectionPrincipalFactor</code> with <code class="func">RZMSNormalization</code> (<a href="chap6.html#X870210EA7912B52A">6.5-6</a>) or <code class="func">RMSNormalization</code> (<a href="chap6.html#X80DE617E841E5BA0">6.5-7</a>) as appropriate.

See also <code class="func">PrincipalFactor</code> (<a href="chap10.html#X86C6D777847AAEC7">10.4-8</a>).

<div class="example"><pre>
gap> S := InverseSemigroup(
> PartialPerm([1, 2, 3, 6, 8, 10],
> [2, 6, 7, 9, 1, 5]),
> PartialPerm([1, 2, 3, 4, 6, 7, 8, 10],
> [3, 8, 1, 9, 4, 10, 5, 6]));;
gap> x := PartialPerm([1, 2, 5, 6, 7, 9],
> [1, 2, 5, 6, 7, 9]);;
gap> D := GreensDClassOfElement(S, x);
<Green's D-class: <identity partial perm on [ 1, 2, 5, 6, 7, 9 ]>>
gap> R := Range(InjectionPrincipalFactor(D));
<Rees 0-matrix semigroup 3x3 over Group(())>
gap> MatrixOfReesZeroMatrixSemigroup(R);
[ [ (), 0, 0 ], [ 0, (), 0 ], [ 0, 0, () ] ]
gap> Size(R);
10
gap> Size(D);
9
gap> S := Semigroup(
> Bipartition([[1, 2, 3, -3, -5], [4], [5, -2], [-1, -4]]),
> Bipartition([[1, 3, 5], [2, 4, -3], [-1, -2, -4, -5]]),
> Bipartition([[1, 5, -2, -4], [2, 3, 4, -1, -5], [-3]]),
> Bipartition([[1, 5, -1, -2, -3], [2, 4, -4], [3, -5]]));;
gap> D := GreensDClassOfElement(S,
> Bipartition([[1, 5, -2, -4], [2, 3, 4, -1, -5], [-3]]));
<Green's D-class: <bipartition: [ 1, 5, -2, -4 ], [ 2, 3, 4, -1, -5 ]
 , [ -3 ]>>
gap> InjectionNormalizedPrincipalFactor(D);
MappingByFunction( <Green's D-class: <bipartition: [ 1, 5, -2, -4 ],
 [ 2, 3, 4, -1, -5 ], [ -3 ]>>, <Rees matrix semigroup 1x1 over
 Group([ (1,2) ])>, function( x ) ... end, function( x ) ... end )</pre></div>

<a id="X86C6D777847AAEC7" name="X86C6D777847AAEC7"></a>

<h5>10.4-8 PrincipalFactor</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrincipalFactor</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalizedPrincipalFactor</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A Rees (0-)matrix semigroup.

If <var class="Arg">D</var> is a D-class of semigroup, then <code class="code">PrincipalFactor(<var class="Arg">D</var>)</code> is just shorthand for <code class="code">Range(InjectionPrincipalFactor(<var class="Arg">D</var>))</code>, and <code class="code">NormalizedPrincipalFactor(<var class="Arg">D</var>)</code> is shorthand for <code class="code">Range(InjectionNormalizedPrincipalFactor(<var class="Arg">D</var>))</code>.

See <code class="func">InjectionPrincipalFactor</code> (<a href="chap10.html#X7EBB4F1981CC2AE9">10.4-7</a>) and <code class="func">InjectionNormalizedPrincipalFactor</code> (<a href="chap10.html#X7EBB4F1981CC2AE9">10.4-7</a>) for more details.

<div class="example"><pre>
gap> S := Semigroup([PartialPerm([1, 2, 3], [1, 3, 4]),
> PartialPerm([1, 2, 3], [2, 5, 3]),
> PartialPerm([1, 2, 3, 4], [2, 4, 1, 5]),
> PartialPerm([1, 3, 5], [5, 1, 3])]);;
gap> PrincipalFactor(MinimalDClass(S));
<Rees matrix semigroup 1x1 over Group(())>
gap> MultiplicativeZero(S);
<empty partial perm>
gap> S := Semigroup(
> Bipartition([[1, 2, 3, 4, 5, -1, -3], [-2, -5], [-4]]),
> Bipartition([[1, -5], [2, 3, 4, 5, -1, -3], [-2, -4]]),
> Bipartition([[1, 5, -4], [2, 4, -1, -5], [3, -2, -3]]));;
gap> D := MinimalDClass(S);
<Green's D-class: <bipartition: [ 1, 2, 3, 4, 5, -1, -3 ],
 [ -2, -5 ], [ -4 ]>>
gap> NormalizedPrincipalFactor(D);
<Rees matrix semigroup 1x5 over Group(())></pre></div>

<a id="X802E2BC9828341A2" name="X802E2BC9828341A2"></a>

<h4>10.5 
 Operations for Green's relations and classes
 </h4>

In this section, we describe some operations related to Green's classes that are available in the Semigroups package.

<a id="X7EDE3F03879B2B12" name="X7EDE3F03879B2B12"></a>

<h5>10.5-1 LeftGreensMultiplier</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftGreensMultiplier</code>( <var class="Arg">S</var>, <var class="Arg">a</var>, <var class="Arg">b</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightGreensMultiplier</code>( <var class="Arg">S</var>, <var class="Arg">a</var>, <var class="Arg">b</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: An element.

If <var class="Arg">S</var> is a semigroup, and <var class="Arg">a</var> and <var class="Arg">b</var> are L-related elements of <var class="Arg">S</var>, then <code class="code">LeftGreensMultiplier</code> returns an element <code class="code">s</code> such that <code class="code">s * <var class="Arg">a</var> = <var class="Arg">b</var></code>. The element <code class="code">s</code> is of the same type as the elements of <var class="Arg">S</var> but may or may not be an element of <var class="Arg">S</var>. In particular, if <var class="Arg">S</var> is not a monoid and <code class="code"><var class="Arg">a</var> = <var class="Arg">b</var></code>, then <code class="code">One(GeneratorsOfSemigroup(S))</code> or an adjoined identity may be returned. Even if <code class="code"><var class="Arg">a</var> <> <var class="Arg">b</var></code>, then it is not guaranteed that the returned element <code class="code">s</code> will belong to <var class="Arg">S</var>. It is guaranteed that the left action of <code class="code">s</code> on the elements of the L-class of <var class="Arg">a</var> is the same as the left action of an element of <var class="Arg">S</var> with the identity adjoined.

<code class="code">LeftGreensMultiplier</code> gives an error if <var class="Arg">a</var> and <var class="Arg">b</var> are not L-related elements of <var class="Arg">S</var>.

The operation <code class="code">RightGreensMultiplier</code> is defined analogously.

<div class="example"><pre>
gap> S := Semigroup(Transformation([4, 4, 3, 5, 3]),
> Transformation([5, 1, 1, 4, 1]),
> Transformation([5, 5, 4, 4, 5]));;
gap> a := Transformation([5, 5, 4, 4, 5]);;
gap> LeftGreensMultiplier(S, a, a);
Transformation( [ 1, 1, 3, 3, 1 ] )
gap> RightGreensMultiplier(S, a, a);
Transformation( [ 5, 5, 5, 4, 5 ] )
gap> b := Transformation([5, 4, 4, 5, 4]);
Transformation( [ 5, 4, 4, 5, 4 ] )
gap> s := LeftGreensMultiplier(S, a, b);
Transformation( [ 1, 3, 3, 1, 3 ] )
gap> s * a;
Transformation( [ 5, 4, 4, 5, 4 ] )
gap> b := Transformation([4, 4, 5, 5, 4]);
Transformation( [ 4, 4, 5, 5, 4 ] )
gap> s := RightGreensMultiplier(S, a, b);
Transformation( [ 4, 4, 4, 5, 4 ] )
gap> a * s = b;
true</pre></div>

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