<p =GAPprompt;>span"(S;/>
3/><div
\begin{array}{ccccccccc}
1&2amp&;&n-1&; n\
&1amp2amp\&;&java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
\{}right \qquad
\left>
{}ccccccccc}
1;2;cdotsampi-1& iamp i+&;i+amp;\codeclassfunc>82 java.lang.StringIndexOutOfBoundsException: Range [128, 127) out of bounds for length 234
&class"table""width"nbspnbsp<<=.76-ReflexiveBooleanMatMonoid>pan
;2ampcdots&;&;i+;&;i+ampcdots
="< />&;/span>< hrefhrefhref"chap7_mjhtml">.-5GossipMonoid<
{rray\right
\]</p>
<p>where <span class=
</dd>< class=M">OrderEndomorphisms(var class"">n)/>
<dt><strong class="Mark"><code class="code">SingularOrderEndomorphisms(<var class="Arg">n</var>)</code></strong></dt>
<dd><p><code class="code">SingularOrderEndomorphisms(<var class="Arg">n</var>)</code> returns the ideal of <code class="code">OrderEndomorphisms(<var class="Arg">n</var>)</code> consisting of the non-invertible elements, when <var class="Arg">n</var> is at least <code class="code">2</code>. The only invertible element in <code class="code">OrderEndomorphisms(<var class="Arg">n</var>)</code> is the identity transformation. Therefore <code class="code">SingularOrderEndomorphisms(<var class="Arg">n</var>)</code> has <span class="SimpleMath">\({2n-1\choose n-1} - 1\)</span> elements.</p>
<p>An endomorphismofvar=Arg</> ahomomorphism classDcode =://.github/chap7_mjX85E9B019877AD7FE =":>EndomorphismMonoid</code>, called with a single argument
<p>If the <var class="Arg">&t; transformation of 0with>
<ul
>> of of the vertices class<>, < class"code class=""[]/> the ofvertex"i/>
</li>
<li><p
</li>
</ul>
<p>See also <code class="func">GeneratorsOfEndomorphismMonoid</code> (<a href="https://gap-packages.github.io/io/doc/chap7_mj.html#X7E93B268823F6478"><span class="RefLink">Digraphs: GeneratorsOfEndomorphismMonoid</span></a>). Note that the performance of <code class="code">EndomorphismMonoid</code> may differ from that of <code class="func">GeneratorsOfEndomorphismMonoid</code> (<a href="https://gap-packages.github.io/io/doc/chap7_mj.html#X7E93B268823F6478"><span class="<transformation monoid of degree 7 withwith 12 generatorsgt;
<h4>7.2 <span class="Heading">
Semigroups of partial permutations
</span></h4>
<p>In this section, we describe the operations in <strong class="pkg">Semigroups</strong> that can be used to create semigroups of partial permutations belonging to several standard classes of example. See <a href="../../../doc/ref/chap54_mj.html#X7D6495F77B8A77BD"><span class="RefLink">Reference: Partial permutations</span></a> for more information about partial permutations.</p>
<div class="func"><table class="func" width=java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 163
<Returns: Munnof.java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
p < ="">/ar is a semilattice, then <ode="code"MunnSemigroupcodereturns inversejava.lang.StringIndexOutOfBoundsException: Range [135, 134) out of bounds for length 280
<p>This function was written jointly by J. D. Mitchell, Yann Péresse (St Andrews), Yanhui Wang (York).</p>
<dl>
<dt><strong class="Mark"><code class="code">POI(<var class="Arg">n</var>)</code></strong></dt>
<dd><p><code class="code">POI(<var class="Arg">n</var>)</code> returns the inverse monoid of partial permutations that preserve the usual order on <span class="SimpleMath">\(\{1, 2, \ldots, n\}\)</span>, where <var class="Arg">n</var> is a positive integer. <code class="code">POI(<var class="Arg">n</var>)</code> is generated by the <span class="SimpleMath">\(\textit{n}\)</span> partial permutations:</p>
<p class="center">\[
left
\{}{cccccjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
1&2&3&\cdots&java.lang.StringIndexOutOfBoundsException: Range [0, 38) out of bounds for length 0
\{}right,\java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
\left(
\begin{array}{ccccccccc}
1&2&\cdots&i-1& i& i+1&i+2&\cdots
&n\\
1&2&\cdots&i-1& i+1&-&i+2&\cdots&n\\
\end{array}\right)
\]</p>
<p>and has <span class="SimpleMath">\(1+\frac{n}{2}{2n\choose n}\)</span> elements.</p>
</divclass="func"< classfunc width10%t>< "">code="">&82;PORI<code>(< class="rg"n</>)/d>td class="dright"&operationnbsp)td<>/ablediv>
<dt< classMarkcode classcode>PORI(< class=rgn<var)</ode/></>
<dd><p><code class="code">PORI(<var class="Arg">n</var>)</code> returns the inverse monoid of partial permutations that preserve or reverse the orientation of <span class="SimpleMath">\(\{1, 2, \ldots, n\}\)</span>, where <span class="SimpleMath">\(n\)</span> is a positive integer. <code class="code">PORI(<var class="Arg">n</var>)</code> is generated by the generators of <code class="code">POPI(<var class="Arg">n</var>)</code>, alongjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</dd>
</dl>
<div class="example"><pre
<span class="GAPprompt">gap>&;&cdotsampn-1
<inverse partial perm java.lang.StringIndexOutOfBoundsException: Range [8, 31) out of bounds for length 14
<span class="GAPprompt">p>here class>\(1 dots \)<panandhas span=SimpleMath>{nchoosen\)</spanelements/java.lang.StringIndexOutOfBoundsException: Index 138 out of bounds for length 138
<inverse partial perm monoid of rank 10 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(S) = 1 + 5 * Binomial(20, 10);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">S := PODI(10);</span>
<inverse partial perm monoid of rank 10 with 11 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := POI(10);</span>
<inverse partial perm monoid of rank 10 with 10 generators>
<span class="<>
monoidthat <class"\\12 ldots java.lang.StringIndexOutOfBoundsException: Range [204, 203) out of bounds for length 671
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubsemigroup(PORI(10), PODI(10))</span>
<span class="GAPprompt">></span> <span class="GAPinput">and IsSubsemigroup(PORI(10), POPI(10))</>
<span class="GAPprompt">></span> <span class="GAPinput">and IsSubsemigroup(PODI(10), POI(10))</span>
<span class="GAPprompt">></span> <span class="GAPinput">and IsSubsemigroup(POPI(10), POI(10));</span>
true</pre<divjava.lang.StringIndexOutOfBoundsException: Range [16, 17) out of bounds for length 16
̻ SingularBrauerMonoid/code( var class="Arg>n/>)/>tdclasstdright> operation&;<td<tr></table</>
<p>Returns: A bipartition monoid.</p>
<p>If <var class="Arg">n</var> is a non-negative integer, then this operation returns the Brauer monoid of degree <var class="Arg">n</var>. The <em>Brauer monoid</em> is the submonoid of the partition monoid consisting of those bipartitions where the size of every block is 2.</p>
<p><code class="code">PartialBrauerMonoid</code> returns the partial Brauer monoid, which is the submonoid of the partition monoid consisting of those bipartitions where the size of every block is <em>at most</em> 2. The partial Brauer monoid contains the Brauer monoid as a submonoid.<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class span">>java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 90
lt; *monoidof degree4with>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubsemigroup(S, JonesMonoid(4));</span>
true
<span class="GAPprompt">gap
10java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
<span class="GAPprompt"> >
< bipartition- ideal 8with>
<span class="GAPprompt">gap14
< * 3 &;
<span class="GAPprompt">gapltregularbipartition *-semigroup of generator&tjava.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<span 14</>/iv
7><div
<p>If <varp>< class""></code thepartial Brauer monoid which is submonoidof partition consistingofthose where sizeof every block isemat most/>2 partial monoidcontainstheBrauer monoid a submonoid</p>
<p><code class="code">SingularJonesMonoid</code>
<iv="example>>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := JonesMonoid(4);</span>
<regular bipartition *-monoid of degree 4 with 3 generators>
<span &tregular bipartition-onoid of 4 with generatorsgt
true
<rue
lt bipartition of8 with generator&;/re/div
<> <var="">n</ar is non-negative integer <code ="code">PartialJonesMonoid/>returnsthepartialJones of <var=Argn/>.Theem monoidem asubsemigroup the Brauermonoid element the Brauer is inthe Jones if partition it isfiner than partition by element the monoid other, anelement thepartialJones monoid be from elementcodeclass"code> by singleton blocks >n</var> is strictly contained in the Motzkin monoid of the same degree.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code classpIf< classArg<> isa non-negative integer then codeclass">Arg"artial Jones monoid/m>isasubsemigroup of the monoid monoidmonoid cin Jonesmonoid ifthepartitionthat finerthanthepartition by elementoftheJones .Inotherwords thepartialJonesmonoid some element< =code>by singleton blocks codeclass=code"[a] [b]/><p>
<R: Abipartition.</>
<p>If <var
<p>Note that the Motzkin monoid of degree <var class="Arg">n</var> contains the partial Jones monoid of degree <var class="Arg">n</var>, but in general, these monoids are not equal; see <code class="func">PartialJonesMonoid</code> (<a href=
<div class="func"><table class="func14
< =func<able ="" ="0%><>dleft<code class="">82;DualSymmetricInverseMonoid/> < classArg>/>)<>td=tdright(nbspoperation )/></r>/></>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"1<pre<divjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
d =f"< ="func=10"
"< ="func>#22;PartialDualSymmetricInverseMonoid( <arclass")/>td class="dright(nbsp;operation&bsp)/></tr<table<divjava.lang.StringIndexOutOfBoundsException: Index 241 out of bounds for length 241
<p>Returns: An inverse bipartition monoid.</p>
<p>If <var class="Arg">n</var> is a java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 37
<p><code class="code">java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 0
><codeclass=func"̶( Arg></ar </d>d classtdright>nbspoperationnbsp<td<tr<table<div>
p> java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 46
<p>If <var class="Arg">n</var> is a positive integer, then this operation returns the uniform block bijection monoid of degree <var class="Arg">n</var>. The <em>uniform block bijection monoid</em> is the submonoid of the partition monoid consisting of the block bijections of degree <span class="SimpleMath">\(n\)</span> where the number of positive integers in a block equals the number of negative integers in that block. The uniform block bijection monoid is also referred to as the <em>factorisable dual symmetric inverse monoid</em>.</p>
<p><code class="code">SingularUniformBlockBijectionMonoid</code> returns the ideal of the uniform block bijection monoid consisting of the non-invertible elements (i.e. those not in the group of units), when <var class="Arg">n</var> is at least 2.</p>
<p><code class="code">PlanarUniformBlockBijectionMonoid</code> returns the submonoid of the uniform block bijection monoid consisting<iv class="func"><table class""=10"tr>tdclass""< =func>#827 < class=rg>tdclass=tdright>&;operation )/>
<p><code class="code">SingularPlanarUniformBlockBijectionMonoid</code> returns the ideal of the planar uniform block bijection monoid consisting of the non-invertible elements (i.e. those not in the group of units), when <var class="Arg">n</var> is at least 2.</p> divclass="func>table=" =10%<trtd=dleft class">#27 n<>)/td< ="nbspoperation&)/ddiv>
<p><code class: An /java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
<div=""><>
<ivfunc>tablefunc=10"<>codeclass"">#27 < class"">/> </>td="">&;operation;)/>/>/>/>
<p>Returns: A bipartition monoid.</p>
<p>If <var class="Arg">n</var> is a positive integer, then this operation returns the planar partition monoid of degree <var class="Arg">n</var> which is the monoid consisting of all the planar bipartitions of degree <var class="Arg">n</var> (planar bipartitions are definedltinverse bijectionmonoid of 4withgenerators&;
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