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Green 's relations
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<div class="ChapSects"><a href="chap10_mj.html#X80C6C718801855E9">10 <span class="Heading">
Green's relations
</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X788D6753849BAD7C">10.1 <span class="Heading">
Creating Green's classes and representatives
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X87558FEF805D24E1">10.1-1 <span class="Heading">XClassOfYClass</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X81B7AD4C7C552867">10.1-2 <span class="Heading">GreensXClassOfElement</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7B44317786571F8B">10.1-3 <span class="Heading">GreensXClassOfElementNC</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7D51218A80234DE5">10.1-4 <span class="Heading">GreensXClasses</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X865387A87FAAC395">10.1-5 <span class="Heading">XClassReps</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X81E5A04F7DA3A1E1">10.1-6 MinimalDClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X834172F4787A565B">10.1-7 <span class="Heading">MaximalXClasses</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7AA3F0A77D0043FB">10.1-8 NrRegularDClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7E45FD9F7BADDFBD">10.1-9 <span class="Heading">NrXClasses</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8140814084748101">10.1-10 <span class="Heading">PartialOrderOfXClasses</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X83B0EDA57F1D2F97">10.1-11 LengthOfLongestDClassChain</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7E872C5381D0DD8A">10.1-12 IsGreensDGreaterThanFunc</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X819CCBD67FD27115">10.2 <span class="Heading">
Iterators and enumerators of classes and representatives
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8566F84A7F6D4193">10.2-1 <span class="Heading">IteratorOfXClassReps</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X867D7B8982915960">10.2-2 <span class="Heading">IteratorOfXClasses</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X820EF2BA7D5D53B4">10.3 <span class="Heading">
Properties of Green's classes
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X85F30ACF86C3A733">10.3-1 <span class="Heading">Less than for Green's classes</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X859DD1C079C80DCC">10.3-2 IsRegularGreensClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7E9BD34B8021045A">10.3-3 IsGreensClassNC</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X855723B17D4AAF8F">10.4 <span class="Heading">
Attributes of Green's classes
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8723756387DD4C0F">10.4-1 GroupHClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X84F1321E8217D2A8">10.4-2 SchutzenbergerGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X81202126806443F9">10.4-3 StructureDescriptionSchutzenbergerGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X838F43FE79A8C678">10.4-4 StructureDescriptionMaximalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X8459E4067C5773AD">10.4-5 MultiplicativeNeutralElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X85B34FFB82C83127">10.4-6 StructureDescription</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7EBB4F1981CC2AE9">10.4-7 InjectionPrincipalFactor</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X86C6D777847AAEC7">10.4-8 PrincipalFactor</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X802E2BC9828341A2">10.5 <span class="Heading">
Operations for Green's relations and classes
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap10_mj.html#X7EDE3F03879B2B12">10.5-1 LeftGreensMultiplier</a></span>
</div></div>
</div>
<h3>10 <span class="Heading">
Green's relations
</span></h3>
<p>In this chapter we describe the functions in <strong class="pkg">Semigroups</strong> for computing Green's classes and related properties of semigroups.</p>
<p><a id="X788D6753849BAD7C" name="X788D6753849BAD7C"></a></p>
<h4>10.1 <span class="Heading">
Creating Green's classes and representatives
</span></h4>
<p>In this section, we describe the methods in the <strong class="pkg">Semigroups</strong> package for creating Green's classes.</p>
<p><a id="X87558FEF805D24E1" name="X87558FEF805D24E1"></a></p>
<h5>10.1-1 <span class="Heading">XClassOfYClass</span></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>
<p>Returns: A Green's class.</p>
<p><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>.</p>
<p>Note that if it is not known to <strong class="pkg">GAP</strong> 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.</p>
<p>The same result can be produced using:</p>
<div class="example"><pre>First(GreensXClasses(S), x -> Representative(x) in class);</pre></div>
<p>but this might be substantially slower. Note that <code class="code">XClassOfYClass</code> is also likely to be faster than</p>
<div class="example"><pre>GreensXClassOfElement(S, Representative(class));</pre></div>
<p><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"><span class="RefLink">Reference: GreensDClassOfElement</span></a>) and <code class="func">GreensDClassOfElementNC</code> (<a href="chap10_mj.html#X7B44317786571F8B"><span class="RefLink">10.1-3</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup(Transformation([1, 3, 2]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([2, 1, 3]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([3, 2, 1]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([1, 3, 1]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := GreensRClassOfElement(S, Transformation([3, 2, 1]));</span>
<Green's R-class: Transformation( [ 3, 2, 1 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">DClassOfRClass(R);</span>
<Green's D-class: Transformation( [ 3, 2, 1 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGreensDClass(DClassOfRClass(R));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">S := InverseSemigroup(</span>
<span class="GAPprompt">></span> <span class="GAPinput">PartialPerm([2, 6, 7, 0, 0, 9, 0, 1, 0, 5]),</span>
<span class="GAPprompt">></span> <span class="GAPinput">PartialPerm([3, 8, 1, 9, 0, 4, 10, 5, 0, 6]));</span>
<inverse partial perm semigroup of rank 10 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := S.1;</span>
[3,7][8,1,2,6,9][10,5]
<span class="GAPprompt">gap></span> <span class="GAPinput">H := HClass(S, x);</span>
<Green's H-class: [3,7][8,1,2,6,9][10,5]>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := RClassOfHClass(H);</span>
<Green's R-class: [3,7][8,1,2,6,9][10,5]>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := LClass(H);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L = LClass(S, PartialPerm([1, 2, 0, 0, 5, 6, 7, 0, 9]));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">DClass(R) = DClass(L);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">DClass(H) = DClass(L);</span>
true</pre></div>
<p><a id="X81B7AD4C7C552867" name="X81B7AD4C7C552867"></a></p>
<h5>10.1-2 <span class="Heading">GreensXClassOfElement</span></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>
<p>Returns: A Green's class.</p>
<p>These functions produce essentially the same output as the <strong class="pkg">GAP</strong> library functions with the same names; see <code class="func">GreensDClassOfElement</code> (<a href="../../../doc/ref/chap51_mj.html#X87C75A9D86122D93"><span class="RefLink">Reference: GreensDClassOfElement</span></a>). The main difference is that these functions can be applied to a wider class of objects:</p>
<dl>
<dt><strong class="Mark"><code class="code">GreensDClassOfElement</code> and <code class="code">DClass</code></strong></dt>
<dd><p><var class="Arg">X</var> must be a semigroup.</p>
</dd>
<dt><strong class="Mark"><code class="code">GreensHClassOfElement</code> and <code class="code">HClass</code></strong></dt>
<dd><p><var class="Arg">X</var> can be a semigroup, \(\mathscr{R}\)-class, \(\mathscr{L}\)-class, or \(\mathscr{D}\)-class.</p>
</dd>
<dd><p>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 \(\mathscr{H}\)-class in row <var class="Arg">i</var> and column <var class="Arg">j</var>.</p>
</dd>
<dt><strong class="Mark"><code class="code">GreensLClassOfElement</code> and <code class="code">LClass</code></strong></dt>
<dd><p><var class="Arg">X</var> can be a semigroup or \(\mathscr{D}\)-class.</p>
</dd>
<dt><strong class="Mark"><code class="code">GreensRClassOfElement</code> and <code class="code">RClass</code></strong></dt>
<dd><p><var class="Arg">X</var> can be a semigroup or \(\mathscr{D}\)-class.</p>
</dd>
</dl>
<p>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.</p>
<p><a id="X7B44317786571F8B" name="X7B44317786571F8B"></a></p>
<h5>10.1-3 <span class="Heading">GreensXClassOfElementNC</span></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>
<p>Returns: A Green's class.</p>
<p>These functions are essentially the same as <code class="func">GreensDClassOfElement</code> (<a href="chap10_mj.html#X81B7AD4C7C552867"><span class="RefLink">10.1-2</span></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 <strong class="pkg">GAP</strong> if <var class="Arg">f</var> is an element of <var class="Arg">X</var>, then no further attempt to verify this is made.</p>
<p>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.</p>
<p>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 <em>a priori</em> 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.</p>
<p>For example, if</p>
<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>
<p>in the semigroup <var class="Arg">X</var> of order-preserving mappings on 20 points, then</p>
<div class="example"><pre>GreensRClassOfElementNC(X, x);</pre></div>
<p>returns an answer relatively quickly, whereas</p>
<div class="example"><pre>GreensRClassOfElement(X, x)</pre></div>
<p>can take a significant amount of time to return a value.</p>
<p>See also <code class="func">GreensRClassOfElement</code> (<a href="../../../doc/ref/chap51_mj.html#X87C75A9D86122D93"><span class="RefLink">Reference: GreensRClassOfElement</span></a>) and <code class="func">RClassOfHClass</code> (<a href="chap10_mj.html#X87558FEF805D24E1"><span class="RefLink">10.1-1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := RandomSemigroup(IsTransformationSemigroup, 2, 1000);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := [1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := EvaluateWord(Generators(S), x);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := GreensRClassOfElementNC(S, x);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(R);</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">L := GreensLClassOfElementNC(S, x);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(L);</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">x := PartialPerm([2, 3, 4, 5, 0, 0, 6, 8, 10, 11]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := LClass(POI(11), x);</span>
<Green's L-class: [1,2,3,4,5][7,6][9,10,11](8)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(L);</span>
165</pre></div>
<p><a id="X7D51218A80234DE5" name="X7D51218A80234DE5"></a></p>
<h5>10.1-4 <span class="Heading">GreensXClasses</span></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>
<p>Returns: A list of Green's classes.</p>
<p>These functions produce essentially the same output as the <strong class="pkg">GAP</strong> library functions with the same names; see <code class="func">GreensDClasses</code> (<a href="../../../doc/ref/chap51_mj.html#X844D20467A644811"><span class="RefLink">Reference: GreensDClasses</span></a>). The main difference is that these functions can be applied to a wider class of objects:</p>
<dl>
<dt><strong class="Mark"><code class="code">GreensDClasses</code> and <code class="code">DClasses</code></strong></dt>
<dd><p><var class="Arg">X</var> should be a semigroup.</p>
</dd>
<dt><strong class="Mark"><code class="code">GreensHClasses</code> and <code class="code">HClasses</code></strong></dt>
<dd><p><var class="Arg">X</var> can be a semigroup, \(\mathscr{R}\)-class, \(\mathscr{L}\)-class, or \(\mathscr{D}\)-class.</p>
</dd>
<dt><strong class="Mark"><code class="code">GreensLClasses</code> and <code class="code">LClasses</code></strong></dt>
<dd><p><var class="Arg">X</var> can be a semigroup or \(\mathscr{D}\)-class.</p>
</dd>
<dt><strong class="Mark"><code class="code">GreensRClasses</code> and <code class="code">RClasses</code></strong></dt>
<dd><p><var class="Arg">X</var> can be a semigroup or \(\mathscr{D}\)-class.</p>
</dd>
</dl>
<p>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.</p>
<p>See also <code class="func">DClassReps</code> (<a href="chap10_mj.html#X865387A87FAAC395"><span class="RefLink">10.1-5</span></a>), <code class="func">IteratorOfDClassReps</code> (<a href="chap10_mj.html#X8566F84A7F6D4193"><span class="RefLink">10.2-1</span></a>), <code class="func">IteratorOfDClasses</code> (<a href="chap10_mj.html#X867D7B8982915960"><span class="RefLink">10.2-2</span></a>), and <code class="func">NrDClasses</code> (<a href="chap10_mj.html#X7E45FD9F7BADDFBD"><span class="RefLink">10.1-9</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup(Transformation([3, 4, 4, 4]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([4, 3, 1, 2]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GreensDClasses(S);</span>
[ <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 ] )> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GreensRClasses(S);</span>
[ <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 ] )> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">D := GreensDClasses(S)[1];</span>
<Green's D-class: Transformation( [ 3, 4, 4, 4 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">GreensLClasses(D);</span>
[ <Green's L-class: Transformation( [ 3, 4, 4, 4 ] )>,
<Green's L-class: Transformation( [ 1, 2, 2, 2 ] )> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GreensRClasses(D);</span>
[ <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 ] )> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">R := GreensRClasses(D)[1];</span>
<Green's R-class: Transformation( [ 3, 4, 4, 4 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">GreensHClasses(R);</span>
[ <Green's H-class: Transformation( [ 3, 4, 4, 4 ] )>,
<Green's H-class: Transformation( [ 1, 2, 2, 2 ] )> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">S := InverseSemigroup([</span>
<span class="GAPprompt">></span> <span class="GAPinput">PartialPerm([2, 4, 1]), PartialPerm([3, 0, 4, 1])]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GreensDClasses(S);</span>
[ <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>> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GreensLClasses(S);</span>
[ <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>> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">D := GreensDClasses(S)[3];</span>
<Green's D-class: <identity partial perm on [ 1, 3 ]>>
<span class="GAPprompt">gap></span> <span class="GAPinput">GreensLClasses(D);</span>
[ <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)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GreensRClasses(D);</span>
[ <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>
<p><a id="X865387A87FAAC395" name="X865387A87FAAC395"></a></p>
<h5>10.1-5 <span class="Heading">XClassReps</span></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>
<p>Returns: A list of representatives.</p>
<p><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, \(\mathscr{D}\)-, \(\mathscr{L}\)-, or \(\mathscr{R}\)-class where appropriate.</p>
<p>The same output can be obtained by calling, for example:</p>
<div class="example"><pre>List(GreensXClasses(obj), Representative);</pre></div>
<p>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.</p>
<p>See also <code class="func">GreensDClasses</code> (<a href="chap10_mj.html#X7D51218A80234DE5"><span class="RefLink">10.1-4</span></a>), <code class="func">IteratorOfDClassReps</code> (<a href="chap10_mj.html#X8566F84A7F6D4193"><span class="RefLink">10.2-1</span></a>), <code class="func">IteratorOfDClasses</code> (<a href="chap10_mj.html#X867D7B8982915960"><span class="RefLink">10.2-2</span></a>), and <code class="func">NrDClasses</code> (<a href="chap10_mj.html#X7E45FD9F7BADDFBD"><span class="RefLink">10.1-9</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup(Transformation([3, 4, 4, 4]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([4, 3, 1, 2]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DClassReps(S);</span>
[ Transformation( [ 3, 4, 4, 4 ] ), Transformation( [ 4, 3, 1, 2 ] ),
Transformation( [ 4, 4, 4, 4 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">LClassReps(S);</span>
[ 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 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">D := GreensDClasses(S)[1];</span>
<Green's D-class: Transformation( [ 3, 4, 4, 4 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">LClassReps(D);</span>
[ Transformation( [ 3, 4, 4, 4 ] ), Transformation( [ 1, 2, 2, 2 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RClassReps(D);</span>
[ Transformation( [ 3, 4, 4, 4 ] ), Transformation( [ 4, 4, 3, 4 ] ),
Transformation( [ 4, 3, 4, 4 ] ), Transformation( [ 4, 4, 4, 3 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">R := GreensRClasses(D)[1];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">HClassReps(R);</span>
[ Transformation( [ 3, 4, 4, 4 ] ), Transformation( [ 1, 2, 2, 2 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">S := SymmetricInverseSemigroup(6);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">e := InverseSemigroup(Idempotents(S));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := MunnSemigroup(e);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := LClassNC(M, PartialPerm([51, 63], [51, 47]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">HClassReps(L);</span>
[ <identity partial perm on [ 47, 51 ]>, [27,47](51), [50,47](51),
[64,47](51), [63,47](51), [59,47](51) ]</pre></div>
<p><a id="X81E5A04F7DA3A1E1" name="X81E5A04F7DA3A1E1"></a></p>
<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>
<p>Returns: The minimal \(\mathscr{D}\)-class of a semigroup.</p>
<p>The minimal ideal of a semigroup is the least ideal with respect to containment. <code class="code">MinimalDClass</code> returns the \(\mathscr{D}\)-class corresponding to the minimal ideal of the semigroup <var class="Arg">S</var>. Equivalently, <code class="code">MinimalDClass</code> returns the minimal \(\mathscr{D}\)-class with respect to the partial order of \(\mathscr{D}\)-classes.</p>
<p>It is significantly easier to find the minimal \(\mathscr{D}\)-class of a semigroup, than to find its \(\mathscr{D}\)-classes.</p>
<p>See also <code class="func">PartialOrderOfDClasses</code> (<a href="chap10_mj.html#X8140814084748101"><span class="RefLink">10.1-10</span></a>), <code class="func">IsGreensLessThanOrEqual</code> (<a href="../../../doc/ref/chap51_mj.html#X7AA204C8850F9070"><span class="RefLink">Reference: IsGreensLessThanOrEqual</span></a>), <code class="func">MinimalIdeal</code> (<a href="chap11_mj.html#X7BC68589879C3BE9"><span class="RefLink">11.8-1</span></a>) and <code class="func">RepresentativeOfMinimalIdeal</code> (<a href="chap11_mj.html#X7CA6744182D07C5B"><span class="RefLink">11.8-2</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">D := MinimalDClass(JonesMonoid(8));</span>
<Green's D-class: <bipartition: [ 1, 2 ], [ 3, 4 ], [ 5, 6 ],
[ 7, 8 ], [ -1, -2 ], [ -3, -4 ], [ -5, -6 ], [ -7, -8 ]>>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := InverseSemigroup(</span>
<span class="GAPprompt">></span> <span class="GAPinput">PartialPerm([1, 2, 3, 5, 7, 8, 9], [2, 6, 9, 1, 5, 3, 8]),</span>
<span class="GAPprompt">></span> <span class="GAPinput">PartialPerm([1, 3, 4, 5, 7, 8, 9], [9, 4, 10, 5, 6, 7, 1]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalDClass(S);</span>
<Green's D-class: <empty partial perm>></pre></div>
<p><a id="X834172F4787A565B" name="X834172F4787A565B"></a></p>
<h5>10.1-7 <span class="Heading">MaximalXClasses</span></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>
<p>Returns: The maximal \(\mathscr{D}\), \(\mathscr{L}\), or \(\mathscr{R}\)-classes of a semigroup.</p>
<p>Let <code class="code">X</code> be one of Green's \(\mathscr{D}\)-, \(\mathscr{L}\)-, or \(\mathscr{R}\)-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.</p>
<p>See also <code class="func">PartialOrderOfDClasses</code> (<a href="chap10_mj.html#X8140814084748101"><span class="RefLink">10.1-10</span></a>), <code class="func">IsGreensLessThanOrEqual</code> (<a href="../../../doc/ref/chap51_mj.html#X7AA204C8850F9070"><span class="RefLink">Reference: IsGreensLessThanOrEqual</span></a>), and <code class="func">MinimalDClass</code> (<a href="chap10_mj.html#X81E5A04F7DA3A1E1"><span class="RefLink">10.1-6</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalDClasses(BrauerMonoid(8));</span>
[ <Green's D-class: <block bijection: [ 1, -1 ], [ 2, -2 ],
[ 3, -3 ], [ 4, -4 ], [ 5, -5 ], [ 6, -6 ], [ 7, -7 ],
[ 8, -8 ]>> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalDClasses(FullTransformationMonoid(5));</span>
[ <Green's D-class: IdentityTransformation> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup(</span>
<span class="GAPprompt">></span> <span class="GAPinput">PartialPerm([1, 2, 3, 4, 5, 6, 7], [3, 8, 1, 4, 5, 6, 7]),</span>
<span class="GAPprompt">></span> <span class="GAPinput">PartialPerm([1, 2, 3, 6, 8], [2, 6, 7, 1, 5]),</span>
<span class="GAPprompt">></span> <span class="GAPinput">PartialPerm([1, 2, 3, 4, 6, 8], [4, 3, 2, 7, 6, 5]),</span>
<span class="GAPprompt">></span> <span class="GAPinput">PartialPerm([1, 2, 4, 5, 6, 7, 8], [7, 1, 4, 2, 5, 6, 3]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalDClasses(S);</span>
[ <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>
<p><a id="X7AA3F0A77D0043FB" name="X7AA3F0A77D0043FB"></a></p>
<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>
<p>Returns: A positive integer, or a list.</p>
<p><code class="code">NrRegularDClasses</code> returns the number of regular \(\mathscr{D}\)-classes of the semigroup <var class="Arg">S</var>.</p>
<p><code class="code">RegularDClasses</code> returns a list of the regular \(\mathscr{D}\)-classes of the semigroup <var class="Arg">S</var>.</p>
<p>See also <code class="func">IsRegularGreensClass</code> (<a href="chap10_mj.html#X859DD1C079C80DCC"><span class="RefLink">10.3-2</span></a>) and <code class="func">IsRegularDClass</code> (<a href="../../../doc/ref/chap51_mj.html#X7F5860927CAD920F"><span class="RefLink">Reference: IsRegularDClass</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup(Transformation([1, 3, 4, 1, 3, 5]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([5, 1, 6, 1, 6, 3]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrRegularDClasses(S);</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">NrDClasses(S);</span>
7
<span class="GAPprompt">gap></span> <span class="GAPinput">AsSet(RegularDClasses(S));</span>
[ <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>
<p><a id="X7E45FD9F7BADDFBD" name="X7E45FD9F7BADDFBD"></a></p>
<h5>10.1-9 <span class="Heading">NrXClasses</span></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>
<p>Returns: A positive integer.</p>
<p><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, \(\mathscr{D}\)-, \(\mathscr{L}\)-, or \(\mathscr{R}\)-class where appropriate. If the actual Green's classes are not required, then it is more efficient to use</p>
<div class="example"><pre>NrHClasses(obj)</pre></div>
<p>than</p>
<div class="example"><pre>Length(HClasses(obj))</pre></div>
<p>since the Green's classes themselves are not created when <code class="code">NrXClasses</code> is called.</p>
<p>See also <code class="func">GreensRClasses</code> (<a href="chap10_mj.html#X7D51218A80234DE5"><span class="RefLink">10.1-4</span></a>), <code class="func">GreensRClasses</code> (<a href="../../../doc/ref/chap51_mj.html#X844D20467A644811"><span class="RefLink">Reference: GreensRClasses</span></a>), <code class="func">IteratorOfRClasses</code> (<a href="chap10_mj.html#X867D7B8982915960"><span class="RefLink">10.2-2</span></a>), and</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([1, 2, 5, 4, 3, 8, 7, 6]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([1, 6, 3, 4, 7, 2, 5, 8]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([2, 1, 6, 7, 8, 3, 4, 5]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([3, 2, 3, 6, 1, 6, 1, 2]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([5, 2, 3, 6, 3, 4, 7, 4]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := Transformation([2, 5, 4, 7, 4, 3, 6, 3]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := RClass(S, x);</span>
<Green's R-class: Transformation( [ 2, 5, 4, 7, 4, 3, 6, 3 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrHClasses(R);</span>
12
<span class="GAPprompt">gap></span> <span class="GAPinput">D := DClass(R);</span>
<Green's D-class: Transformation( [ 2, 5, 4, 7, 4, 3, 6, 3 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrHClasses(D);</span>
72
<span class="GAPprompt">gap></span> <span class="GAPinput">L := LClass(S, x);</span>
<Green's L-class: Transformation( [ 2, 5, 4, 7, 4, 3, 6, 3 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrHClasses(L);</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">NrHClasses(S);</span>
1555
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup(Transformation([4, 6, 5, 2, 1, 3]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([6, 3, 2, 5, 4, 1]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([1, 2, 4, 3, 5, 6]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([3, 5, 6, 1, 2, 3]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([5, 3, 6, 6, 6, 2]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([2, 3, 2, 6, 4, 6]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([2, 1, 2, 2, 2, 4]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([4, 4, 1, 2, 1, 2]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrRClasses(S);</span>
150
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(S);</span>
6342
<span class="GAPprompt">gap></span> <span class="GAPinput">x := Transformation([1, 3, 3, 1, 3, 5]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D := DClass(S, x);</span>
<Green's D-class: Transformation( [ 1, 3, 3, 1, 3, 5 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrRClasses(D);</span>
87
<span class="GAPprompt">gap></span> <span class="GAPinput">S := SymmetricInverseSemigroup(10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrDClasses(S); NrRClasses(S); NrHClasses(S); NrLClasses(S);</span>
11
1024
184756
1024
<span class="GAPprompt">gap></span> <span class="GAPinput">S := POPI(10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrDClasses(S);</span>
11
<span class="GAPprompt">gap></span> <span class="GAPinput">NrRClasses(S);</span>
1024</pre></div>
<p><a id="X8140814084748101" name="X8140814084748101"></a></p>
<h5>10.1-10 <span class="Heading">PartialOrderOfXClasses</span></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>
<p>Returns: A digraph.</p>
<p>Let <code class="code">X</code> be one of Green's \(\mathscr{D}\)-, \(\mathscr{L}\)-, or \(\mathscr{R}\)-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 <strong class="pkg">digraphs</strong> package).</p>
<p>The partial order on the <code class="code">X</code>-classes is defined as follows.</p>
<dl>
<dt><strong class="Mark">Green's \(\mathscr{D}\)-relation:</strong></dt>
<dd><p><span class="SimpleMath">\(x\leq y\)</span> if and only if <span class="SimpleMath">\(S ^ 1xS ^ 1\)</span> is a subset of <span class="SimpleMath">\(S ^ 1yS ^ 1\)</span>.</p>
</dd>
<dt><strong class="Mark">Green's \(\mathscr{L}\)-relation:</strong></dt>
<dd><p><span class="SimpleMath">\(x\leq y\)</span> if and only if <span class="SimpleMath">\(S ^ 1x\)</span> is a subset of <span class="SimpleMath">\(S ^ 1y\)</span>.</p>
</dd>
<dt><strong class="Mark">Green's \(\mathscr{R}\)-relation:</strong></dt>
<dd><p><span class="SimpleMath">\(x\leq y\)</span> if and only if <span class="SimpleMath">\(xS ^ 1\)</span> is a subset of <span class="SimpleMath">\(yS ^ 1\)</span>.</p>
</dd>
</dl>
<p>See also <code class="func">GreensDClasses</code> (<a href="chap10_mj.html#X7D51218A80234DE5"><span class="RefLink">10.1-4</span></a>), <code class="func">GreensDClasses</code> (<a href="../../../doc/ref/chap51_mj.html#X844D20467A644811"><span class="RefLink">Reference: GreensDClasses</span></a>), <code class="func">IsGreensLessThanOrEqual</code> (<a href="../../../doc/ref/chap51_mj.html#X7AA204C8850F9070"><span class="RefLink">Reference: IsGreensLessThanOrEqual</span></a>), and <code class="func">\<</code> (<a href="chap10_mj.html#X85F30ACF86C3A733"><span class="RefLink">10.3-1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup(Transformation([2, 4, 1, 2]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([3, 3, 4, 1]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PartialOrderOfDClasses(S);</span>
<immutable digraph with 4 vertices, 3 edges>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGreensLessThanOrEqual(GreensDClasses(S)[1],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> GreensDClasses(S)[2]);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGreensLessThanOrEqual(GreensDClasses(S)[2],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> GreensDClasses(S)[1]);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGreensLessThanOrEqual(GreensDClasses(S)[3],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> GreensDClasses(S)[1]);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">S := InverseSemigroup(</span>
<span class="GAPprompt">></span> <span class="GAPinput">PartialPerm([1, 2, 3], [1, 3, 4]),</span>
<span class="GAPprompt">></span> <span class="GAPinput">PartialPerm([1, 3, 5], [5, 1, 3]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(S);</span>
58
<span class="GAPprompt">gap></span> <span class="GAPinput">PartialOrderOfDClasses(S);</span>
<immutable digraph with 5 vertices, 4 edges>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGreensLessThanOrEqual(GreensDClasses(S)[1],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> GreensDClasses(S)[2]);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGreensLessThanOrEqual(GreensDClasses(S)[5],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> GreensDClasses(S)[2]);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGreensLessThanOrEqual(GreensDClasses(S)[3],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> GreensDClasses(S)[4]);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGreensLessThanOrEqual(GreensDClasses(S)[4],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> GreensDClasses(S)[3]);</span>
true</pre></div>
<p><a id="X83B0EDA57F1D2F97" name="X83B0EDA57F1D2F97"></a></p>
<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>
<p>Returns: A non-negative integer.</p>
<p>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_mj.html#X8140814084748101"><span class="RefLink">10.1-10</span></a>).</p>
<p>The partial order on the \(\mathscr{D}\)-classes is defined by <span class="SimpleMath">\(x\leq y\)</span> if and only if <span class="SimpleMath">\(S ^ 1xS ^ 1\)</span> is a subset of <span class="SimpleMath">\(S ^ 1yS ^ 1\)</span>. A <em>chain</em> of \(\mathscr{D}\)-classes is a collection of <code class="code">n</code> \(\mathscr{D}\)-classes <span class="SimpleMath">\(D_{1}, D_{2}, \ldots D_{n}\)</span> such that <span class="SimpleMath">\(D_{1} < D_{2} < \cdots < D_{n}\)</span>. The <em>length</em> of such a chain is <code class="code">n - 1</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := TrivialSemigroup();;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LengthOfLongestDClassChain(S);</span>
0
<span class="GAPprompt">gap></span> <span class="GAPinput">T := ZeroSemigroup(5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LengthOfLongestDClassChain(T);</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">U := MonogenicSemigroup(14, 7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LengthOfLongestDClassChain(U);</span>
13
<span class="GAPprompt">gap></span> <span class="GAPinput">V := FullTransformationMonoid(6);</span>
<full transformation monoid of degree 6>
<span class="GAPprompt">gap></span> <span class="GAPinput">LengthOfLongestDClassChain(V);</span>
5</pre></div>
<p><a id="X7E872C5381D0DD8A" name="X7E872C5381D0DD8A"></a></p>
<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>
<p>Returns: A function.</p>
<p><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 \(\mathscr{D}\)-class of <code class="code">x</code> in <var class="Arg">S</var> is greater than or equal to the \(\mathscr{D}\)-class of <code class="code">y</code> in <var class="Arg">S</var> under the usual ordering of Green's \(\mathscr{D}\)-classes of a semigroup.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup(Transformation([1, 3, 4, 1, 3]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([2, 4, 1, 5, 5]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([2, 5, 3, 5, 3]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([5, 5, 1, 1, 3]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">reps := ShallowCopy(AsSet(DClassReps(S)));</span>
[ Transformation( [ 1, 1, 1, 1, 1 ] ),
Transformation( [ 1, 3, 1, 3, 3 ] ),
Transformation( [ 1, 3, 4, 1, 3 ] ),
Transformation( [ 2, 4, 1, 5, 5 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Sort(reps, IsGreensDGreaterThanFunc(S));</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">reps;</span>
[ Transformation( [ 2, 4, 1, 5, 5 ] ),
Transformation( [ 1, 3, 4, 1, 3 ] ),
Transformation( [ 1, 3, 1, 3, 3 ] ),
Transformation( [ 1, 1, 1, 1, 1 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGreensLessThanOrEqual(DClass(S, reps[2]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> DClass(S, reps[1]));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">S := DualSymmetricInverseMonoid(4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGreensDGreaterThanFunc(S)(S.1, S.3);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGreensDGreaterThanFunc(S)(S.3, S.1);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGreensLessThanOrEqual(DClass(S, S.3),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> DClass(S, S.1));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGreensLessThanOrEqual(DClass(S, S.1),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> DClass(S, S.3));</span>
false</pre></div>
<p><a id="X819CCBD67FD27115" name="X819CCBD67FD27115"></a></p>
<h4>10.2 <span class="Heading">
Iterators and enumerators of classes and representatives
</span></h4>
<p>In this section, we describe the methods in the <strong class="pkg">Semigroups</strong> package for incrementally determining Green's classes or their representatives.</p>
<p><a id="X8566F84A7F6D4193" name="X8566F84A7F6D4193"></a></p>
<h5>10.2-1 <span class="Heading">IteratorOfXClassReps</span></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>
<p>Returns: An iterator.</p>
<p>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"><span class="RefLink">Reference: Iterators</span></a> for more information on iterators.</p>
<p>See also <code class="func">GreensRClasses</code> (<a href="../../../doc/ref/chap51_mj.html#X844D20467A644811"><span class="RefLink">Reference: GreensRClasses</span></a>), <code class="func">GreensRClasses</code> (<a href="chap10_mj.html#X7D51218A80234DE5"><span class="RefLink">10.1-4</span></a>), and <code class="func">IteratorOfRClasses</code> (<a href="chap10_mj.html#X867D7B8982915960"><span class="RefLink">10.2-2</span></a>).</p>
<p><a id="X867D7B8982915960" name="X867D7B8982915960"></a></p>
<h5>10.2-2 <span class="Heading">IteratorOfXClasses</span></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>
<p>Returns: An iterator.</p>
<p>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"><span class="RefLink">Reference: Iterators</span></a> for more information on iterators.</p>
<p>This function is useful if you are, for example, looking for an \(\mathscr{R}\)-class of a semigroup with a particular property but do not necessarily want to compute all of the \(\mathscr{R}\)-classes.</p>
<p>See also <code class="func">GreensRClasses</code> (<a href="chap10_mj.html#X7D51218A80234DE5"><span class="RefLink">10.1-4</span></a>), <code class="func">GreensRClasses</code> (<a href="../../../doc/ref/chap51_mj.html#X844D20467A644811"><span class="RefLink">Reference: GreensRClasses</span></a>), and <code class="func">NrRClasses</code> (<a href="chap10_mj.html#X7E45FD9F7BADDFBD"><span class="RefLink">10.1-9</span></a>).</p>
<p>The transformation semigroup in the example below has 25147892 elements but it only takes a fraction of a second to find a non-trivial \(\mathscr{R}\)-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 \(\mathscr{R}\)-class with more than 1000 elements.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens := [Transformation([2, 4, 1, 5, 4, 4, 7, 3, 8, 1]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([3, 2, 8, 8, 4, 4, 8, 6, 5, 7]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([4, 10, 6, 6, 1, 2, 4, 10, 9, 7]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([6, 2, 2, 4, 9, 9, 5, 10, 1, 8]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([6, 4, 1, 6, 6, 8, 9, 6, 2, 2]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([6, 8, 1, 10, 6, 4, 9, 1, 9, 4]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([8, 6, 2, 3, 3, 4, 8, 6, 2, 9]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([9, 1, 2, 8, 1, 5, 9, 9, 9, 5]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([9, 3, 1, 5, 10, 3, 4, 6, 10, 2]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([10, 7, 3, 7, 1, 9, 8, 8, 4, 10])];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup(gens);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">iter := IteratorOfRClasses(S);</span>
<iterator>
<span class="GAPprompt">gap></span> <span class="GAPinput">for R in iter do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> if Size(R) > 1 then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> break;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R;</span>
<Green's R-class: Transformation( [ 6, 4, 1, 6, 6, 8, 9, 6, 2, 2 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(R);</span>
21600
<span class="GAPprompt">gap></span> <span class="GAPinput">S := InverseSemigroup(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PartialPerm([1, 2, 3, 4, 5, 6, 7, 10, 11, 19, 20],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [19, 4, 11, 15, 3, 20, 1, 14, 8, 13, 17]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PartialPerm([1, 2, 3, 4, 6, 7, 8, 14, 15, 16, 17],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [15, 14, 20, 19, 4, 5, 1, 13, 11, 10, 3]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PartialPerm([1, 2, 4, 6, 7, 8, 9, 10, 14, 15, 18],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [7, 2, 17, 10, 1, 19, 9, 3, 11, 16, 18]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PartialPerm([1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [8, 3, 18, 1, 4, 13, 12, 7, 19, 20, 2, 11]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PartialPerm([1, 2, 3, 4, 5, 6, 7, 9, 11, 15, 16, 17, 20],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [7, 17, 13, 4, 6, 9, 18, 10, 11, 19, 5, 2, 8]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PartialPerm([1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 18],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [10, 20, 11, 7, 13, 8, 4, 9, 2, 18, 17, 6, 15]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PartialPerm([1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 17, 18],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [10, 20, 18, 1, 14, 16, 9, 5, 15, 4, 8, 12, 19, 11]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PartialPerm([1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 15, 16, 19, 20],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [13, 6, 1, 2, 11, 7, 16, 18, 9, 10, 4, 14, 15, 5, 17]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PartialPerm([1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 20],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [5, 3, 12, 9, 20, 15, 8, 16, 13, 1, 17, 11, 14, 10, 2]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PartialPerm([1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 13, 17, 18, 19, 20],</ | |
