>n</ar )/td >< class"" >( operation;)</td ><tr /table ></div ngth 64var class="Arg" >n</var > is a integer greater than 1, then this function returns the semigroup of non-invertible transformations, which is generated by the <code class="code" ><var class="iv class=" >table ="" width0"<>tdleft ="&82;PartialOrderEndomorphisms/> var =A" >n<var )td td ="tdright" >( operation )</td ></tr ></table StandardPBR semigroups<span />div class class> classcode OrderEndomorphismsvar ">)> span classdd >> class"" OrderEndomorphisms<var ar"rg" /var >)/dereturns the of that the order <span class=SimpleMath\\1 ,\, n\\<span where classArg/> a positive. code ="code" OrderEndomorphisms classArg<var code generated thespan ="SimpleMath" >(\{n + 1\)/pantransformations/>span class"GAPprompt" >gap>/span <span class="GAPinput" Size)<span div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ OrderEndomorphisms</code >( <var class="Arg" >n</var > )</td ><td class="tdright" >( operation )</td ></tr > matrices"827;SingularOrderEndomorphisms/code>( Argn< /d>td "" >&;&;<td<tr <table div >div ="func>&; <><href"chap7_mj.htmlX78DF50747A28098C" >7..6-ReflexiveBooleanMatMonoida>/pandiv class="func" ><table classspan =ContSS>br /=nocss>;<a href">dl dt strong "ark>>< ="code" > Argvar > or order< ="\,2 ldots, n\)pan,where ar" >var code ="(< class" "n/ar> generatedbythe bijective transformation that the orderon< class=SimpleMath>\(\1 , \, n\\ The monoid>var "n/ar)/code>has < class" ">(2\choose n-1} - n\)/span> elements. "center" >\[dd dt strong ="Mark" code =code >PartialOrderAntiEndomorphisms ="Arg>/>/>/t span "" \(i=0,ldotsn-1\)</span hasspan class=SimpleMath\{n-1choose})/> elements/>dd >dt ><strong class="Mark" ><code class="code" >SingularOrderEndomorphisms(<var class="Arg" >n</var >)</code ></strong ></dt >dd ><p><code class="code" >SingularOrderEndomorphisms(<var <span class="ContSS" ><br />span ="nocss" ><regulartransformationmonoid degree5with 5generatorsgt;dd >dt ><strong class="Mark" ><code class="code" >OrderAntiEndomorphisms(<var class="Arg" >n</var >)</code strong ></dt >gth 4dd >dt ><strong class="Mark" ><code class="code" [IdentityTransformation"rg>nturns amonoid of on >< =">/> + 1/ode> that isi to the consisting of all partial that preserve usual order on \\{1 2, ldots \\ dd class;regulartransformationmonoid ofdegree 5 generators;dt strong =Mark <code ="code" PartialOrderAntiEndomorphisms ="Arg class=" br /span ="ocss" >bsp;nbsp/pan>a href"chap7_mj.X8411EBD97A220921" >7.8-2 MonogenicSemigroup</a></span >"code" >PartialAntiOrderEndomorphisms(<arclass="rg" >n</>/code returnsamonoidof java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4div class>7.16 EndomorphismMonoid/h5span ="" >ap;/pan span ="" S: (5)<>span class>>>< class(S);</span span class="GAPprompt" >gap></span > <span class="GAPinput" >Size(S) = Binomial(2 * 5 - 1, 5 - 1);</span >span class="GAPprompt" >gap></span > <span of < class"" >digraphvar is homomorphism <code class="func" >igraphHomomorphism</code > (<a href=httpsgap-packages.io/io/doc.html #X85E9B019877AD7FE">Digraphs DigraphHomomorphism/span>/a>) from digraph/var> back toitself. span class="GAPprompt" >gap></span > <span class="GAPinput" >SingularOrderEndomorphismsspan class="GAPprompt" >gap></span > <span ul >span class="GAPprompt" >gap&<li<pA list positive integersof size number ofvertices of <var ="Arg" >digraph/var , wherevar class="Arg" >colorsvar < classcode >i]</code >is colourof < =code ><code ./>span java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0span class="GAPprompt" >gap></span > <span class="GAPinput" >V := PartialOrderAntiEndomorphisms(6);</span >span pre ></div >"X868955247F2AFAA5" name="X868955247F2AFAA5" ></a></p>div class="func" ><table class="<transformation monoid of degree 3 with 3 generators> "" ><table class=func="0%>tr> >code "unc>#827 EndomorphismMonoid/code>( digraph var class=Arg> tdright&;operation)/>/>table div span ="" >><span span ="GAPinput" >(gr;/span >"" >>/> s class"" S:=gr1 ,2];<span code classcodeEndomorphismMonoidcode > called a single, returns monoid allendomorphisms <var classArgdigraphvar >.</p>gth 4ul >li ><p>A list of positive integers of size the number of vertices of <var class="java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 0 li >li ><p>A list of lists, such that <var class="Arg" >colors</var ><code class="code" >[i]</code > is a list of all vertices with colour <code class="code" >i</code >.</p>li >ul >code class="func" >GeneratorsOfEndomorphismMonoid</code > (<a href<>Returns:The semigroup a semilattice.</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >gr := Digraph(List([1 .. 3], x -> [1 .. 3]));;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >EndomorphismMonoid(gr);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >gr := CompleteDigraph(3);;<>If <arclassArg></>isa < class></> the semigroup of partial permutations of isomorphisms of principal ideals of <var "Arg" >S</var >; called the <em >Munn semigroup</em > of <var class="Arg" >S</var >.</p>span class="GAPprompt" >gap></java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0span class="GAPprompt" >gap></span > <span class<span classGAPprompt>><span < class"GAPinput> =InverseSemigroup([/spanjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92 span classltpartial semigroup rank0 with2gtspan class="GAPprompt" >gapspan ="GAPprompt" gap;</span span ="GAPinput" > =MunnSemigroupT)<span span class class"GAPprompt" g&;/> span "GAPinput" NrIdempotentsM);/span span class60span class="GAPprompt" >gap>span classGAPpromptgapgt<span <span class">);span> pre ></div >span >Returns: inversemonoid partialpermutations/>section we describetheoperationsin < class="pkg" ></trong thatcanbe tocreatesemigroups of partialpermutationsto several classesexample Seehref=/./.docref/hap54_mj.">RefLinkReference permutations/ moreinformationpartial<pjava.lang.StringIndexOutOfBoundsException: Index 383 out of bounds for length 383span ="GAPprompt" >gap></span > <span class="GAPinput" >S = SymmetricInverseMonoid(4);</span >div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ MunnSemigroup</code >( <var class="Arg" >S</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table true/pre ></iv>var class="Arg" >S</var > is a semilattice, then <code class="code" >MunnSemigroup</code > returns the inverse semigroup of partial permutations of isomorphisms of principal ideals of <var div class"func" ><able class="func" ="10%" >tr td classtdleft><ode="" &82;PODIcode (<class"> tdclass=tdright" > <td </r><table ><>rk).</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <<>: An inversemonoidof partialpermutations related to order/p>span class="GAPprompt" >></span > <span class="GAPinput" >PartialPerm([1, 2, 7, 9], [5, 6, 4, 3])]);</span >span class="GAPprompt" >gap></span > <span classspan class="GAPprompt" >gap&\(span class="GAPprompt" >gap></span > <span class="GAPinput" >NrIdempotents(M);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >NrIdempotents(S);</span >pre ></div >"X82D9619B7845CAEB" name="X82D9619B7845CAEB" ></a></p>div class="func" ><table endarray\) qquad9) out of bounds for length 14code classdiv class="example" >dt < class="Mark" >< ="code" >PODI(< class"Arg" >n/>)/>/></>span =GAPpromptgap;/pan=>:(;span span dd pre ></div ><t><trong="Mark" > =code POPI(< class<>/>>/dt "X85D841AE83DF101C" namediv =func class" width" 0%>tr classtdleft< classfuncW </> var =><var </td < classtdright(nbspoperation&;)</d>/tr >table>div >div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ PODI</code >( <var class="Arg" >n</var > )</td ><td class="tdright" >( operation )</td ></tr ></table &2∓\cdots;n-1∓\java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34table ="" ="0%>tdclass=tdleft>(nbsp;operation /tr>/table> "" ><code ="" >PORI<var ="rg" ><>)<code ><strong /dt java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95dl >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>"center" >\[span class="SimpleMath" >\i = , ldots, n -1\)</>, and has < class=">({n\choose n}span>> elements. dd >dt ><strong class="Mark" ><code class="code" >PODI(<var class="Arg" >n</var >)</code ></strong ></dt >dd <class>ODI<ar=Arg">n) returns the inverse monoid of partial permutations preserve or reverse the usual order onon SimpleMath">(\{, , \ldots,n\}\) , where n is a positive integer. PODI(n ) is generated by the generators of POI(n ) dd >dt true>/>dd ><p><code class="code" >POPI(<var class="Arg" >n</var >)</code > returns the inverse monoid of partial permutations that preserve the orientation of <span class="SimpleMath" >\(\{1,2,\ldots, n\}\)</span >, where <span class="SimpleMath" >\(n\)</span > is a positive integer. <code class="code" >POPI(<var class="Arg" >n</var >)</code > is generated by the partial permutations:</p>"center" >\[dd >dt ><strong class="Mark" ><code <iv class="func" ><table class="func" width=10%><tr >< =tdleft">PartitionMonoidArgn<var <td >td class=";operation ) <>code class"code>ORI(/var>({,2 ldots,n})span,where>(\</>is integer<ode =code "PORIArg><var )<code bythegeneratorsof < "" POPI< ="" ></var )</ode, along the the usualorderon< class"" >\({1, , ldots \\<span .<code =code >PORI< classArg"n )SimpleMath"\(fracn{}2nchoose n}-n(n+1\elementsdd >dl >div classspan "" >gap><span span class">: 1;/span span classGAPinput() 5 *Binomial 1)<span "gap&;/pan < class=" > :=PODI(0;/pan> Index 62 out of bounds for length 62span "" >S ::= POI1);<span span class="GAPprompt" >gap></span > <span class="GAPinput" >Size(S) = Binomial(20, 10);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >IsSubsemigroup(PORI(10), PODI(10))</span >span class="APinput>nd IsSubsemigroupIsSubsemigroup(1), 10) span class="GAPprompt" >></span > <span class="GAPinput" >and IsSubsemigroup(PODI(10), POI(10))</span >ut of bounds for length 66pre ></div >"func" <table classfuncwidth0">tr< class=" "BrauerMonoidn/var> )>nbsp&;<td tr /></>span ="< =" "10" tr class"#2;<>< class"/><>
Semigroups of bipartitions
</
span ></h4>
<p>In this
section , we describe the operations in <
strong class=
"pkg" >Semigroups</
strong > that ca
n be used to create bipartition semigroups belonging to several standard classes of java.lang.StringIndexOutOfBoundsException: Range [0, 188) out of bounds for length 0"X7E4B61FF7CCFD74A" name="X7E4B61FF7CCFD74A" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ PartitionMonoid</code >( <var class="Arg" >n</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >div classfunc< class=unc =00"(nbsp&bsp/n style='color:red'>td>/table ><ivdiv class="func" ><table var class="Arg" >n</var > is<span class="GAPprompt" >gapgt<span <span class="GAPinput" >S := BrauerMonoid(4);</span >code class="code" >SingularPartitionMonoid</code > returns the ideal of the partition monoid consisting of the non-invertible elements (i.e. those not in <regularbipartition- ofdegree with 3 generators;var class="Arg" >n</var > is positive, then <code class="code" >RookPartitionMonoid</code > returns submonoid of thespan class="GAPprompt" >gap></span > <span class="GAPinput" >S := PartitionMonoid(4);</span >span class="GAPprompt" >gap>lt;regular *semigroup of degree 1 generator;span class="GAPprompt" >gap></span > <span class;regular bipartition -monoidofdegree with8generatorsgtspan class="GAPprompt" >gap></span > <span class="GAPinput" >Size(S) - Size(T) = Factorial(4);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >S := RookPartitionMonoid(4);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Size(S);</span >pre </iv>"X79D33B2E7BA3073A" name="X79D33B2E7BA3073A" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ BrauerMonoid</code >( <var class="Arg" >n</var > )</td ><td class="tdright" >( operation )</td 7</pre /iv>div class="func" ><table class classfunc>< class"func =" 10">>code classfunc>#27 </code (< class="" <java.lang.StringIndexOutOfBoundsException: Range [145, 144) out of bounds for length 219"" ><table class=func ="10%" >tr >td class"dleft>func">&827 SingularBrauerMonoid ( n/r> var class="Arg" >n</var > is a non-negative integer, then this operation returns the Brauer < class=func< ="width=" 10"<"< class=func>#27 SingularJonesMonoidArg/> <td td =tdright&;nbsp/d>tr table <div code =code PartialBrauerMonoid>returns partialBrauer,whichis the ofthe monoid bipartitions the ofeveryblock <>at</em .The Brauer Brauer as submonoid.p>div class="example" ><pre >span class= class"span class=span class="GAPprompt" >gap<regular bipartition *-semigroupideal degree 8with 1generator><p></>span class="GAPprompt" >gap></"X8458B0F7874484CE" >/a>/pjava.lang.StringIndexOutOfBoundsException: Range [62, 63) out of bounds for length 62span class="GAPprompt" >gap></span > <span <iv class=">< class=func" width="00%" ><tr "" >< class"&82;PartialJonesMonoid < class" ><)/<td "(nbspoperation&;/> span class="GAPprompt" >gap></span > <span class="GAPinput" >IsSubsemigroup(S, BrauerMonoid(3));</span >span class="GAPprompt" >gap></span > <span class="<>Returns A bipartition monoid. pre ></div >"X8378FC8B840B9706" name="X8378FC8B840B9706" ></a></p>div class="func" ><table class="func" width="100%" ><tr p>If classArg</> anon-negative integer,then class"PartialJonesMonoid><var <>partialJones</>is of partialBrauer . An of partial monoid containedin thepartial monoid the that defines finerthan the defined some of Jones.In wordsan ofthe Jones can formed some < ="" x/>ofthe code class=code >a,b<code <code class"ode" x/>byblocks class"[] b]/./java.lang.StringIndexOutOfBoundsException: Index 729 out of bounds for length 729 div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ SingularJonesMonoid</code >( <var class="Arg" >n</var > )</td ><td class="tdrightjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 "" >n</ar is a non-negative , thenoperationreturns Jonesmonoidof <var ="" ><var .Theem Jonesmonoidem >is subsemigroupthemonoidof are; code classfuncPlanarPartitionMonoid(a =.#"< =" >.-<span />.The is to class"button>Temperley-Lieb monoid. code class="code" >SingularJonesMonoid</code > returns the ideal &;regularbipartition -monoid degree 3generators>"" >pre span ="&t;< " >: (4;span span =" TemperleyLiebMonoid()/java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96 span class="java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 3 "java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for length 0 div class"func>table =" func width=100%"td class=tdleft>func </code > <var class"" >n</ar )</d<d class="tdright" >(&;operationnbsp/d>/r<table <div java.lang.StringIndexOutOfBoundsException: Index 226 out of bounds for length 226var ="" >n<var is a non-negative, then< ="code>artialJonesMonoid/>returnsthe partial Jones degree< =" Arg">n<> em>artial Jonesmonoidis ofthepartialBrauer monoid.Anelementofthe partialBrauer is ontained thepartialJones monoid if the partition itdefinesis defined bysome monoid ,anelementofthe canbeformedfrom some codeclass" "x/>of the Jonesmonoidbyreplacing someblocks< =code>[a, b] of code class=code" <code > by < =code >a],[b<code ./>div class="example" ><pre >span class="GAPprompt" >gap"" <>span class="GAPprompt" >gap> class="APprompt" >gap;/pan<pan =GAPinputS =MotzkinMonoid4;/>span class=GAPprompt>ap>/pan<pan=GAPinput> : PartialJonesMonoid);</panspan class="GAPprompt" gapgt;</pan span class=GAPinput>sSubsemigroup )<span span class=GAPinput"IsSubsemigroup(,T) span class="GAPprompt" >gap></span > <span ="" >&;<s>< ="" >()<span >span class="< name"X83C7587C81B985BA>/>/java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62 span class="GAPprompt" >gap></span <div class"" ><ableclassfuncwidth10"d class=dleft" >code class="func>#827 (varclass=" ">td class=" >&;operation;<td /r<table /iv"X7DB8CB067CBE1254" name="X7DB8CB067CBE1254" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ AnnularJonesMonoid</code >( <var class="Arg" >n</var > )</td ><td class="tdright" >( operation )</td ><ivclass"unc>tableclass=" func" width" 0%>tr class"tdleft>codeclass=" "&827 PartialDualSymmetricInverseMonoid Arg"n/ar > operation&;<td tr >/>/>var class="Arg" >n<div class="example" ><pre span class"GAPprompt" >&;/span span "" >S ()<span >span <p>div classlength 37var > is non-negative integerthen returns Motzkin degree ="" nvar > /> theBrauerofbipartitionsare planararein< "hap3_mj.X7C18DB427C9C0917" < =RefLink>s/./java.lang.StringIndexOutOfBoundsException: Index 393 out of bounds for length 393div class="example" >span class="GAPprompt" >gap></span > <span class="GAPinput" >T := PartialJonesMonoid(4);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >IsSubsemigroup(S, T);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Size(S);</span >span class="GAPpromptdiv =func" ><able"" ="0%" tr >td classtdleft class>22;SingularUniformBlockBijectionMonoidvar class"" >var ><td >td ="" (nbsp; </>/<table >/>pre ></div >"X83C7587C81B985BA" name="div class=" " =" width0%">>tclass=" code ="&827;SingularFactorisableDualSymmetricInverseMonoid < =" </ar<td <=tdrightnbsp&)/d/>table />tr =tdleftcode =func>82 SingularPlanarUniformBlockBijectionMonoid class"n/>)/>< class=" tdright"( operation )<>/div>div class=" div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ DualSymmetricInverseMonoid</code >( <var class="Arg" >n</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >table =func width=0%><tr < =tdleft><odeclass"" ̻SingularDualSymmetricInverseMonoid> <var ="rg" n/>)<td < "" (nbspoperation;<td <tr <table div java.lang.StringIndexOutOfBoundsException: Index 242 out of bounds for length 242"> class" func width"00" >tr >< class="dleft" ><code class="func" >827PartialDualSymmetricInverseMonoid/> <var class=>n<var><td ><td class="tdright" >(&;operationnbsp;<t>/><><div >var class="Arg" >n</var > is a positive integer, then the operations <code class="code" >DualSymmetricInverseSemigroup</code > and <code class="code" >DualSymmetricInverseMonoid</code > return the dual symmetric inverse monoid of degree <var class="Arg" >n</var >, which is the subsemigroup of the partition monoid consisting of the block bijections of degree <var class="Arg" >n</var >.</p>code class="code" >SingularDualSymmetricInverseMonoid</code > returns the ideal of the dual symmetric inverse monoidjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0code class="code" >PartialDualSymmetricInverseMonoid</code > returns the submonoid of the dual symmetric inverse monoid of degree <code class="code" ><var class="Arg" >n</var > + 1</code > consisting of those block bijections with <code class="code" ><var class="Arg" >n</var > + 1</code > and <code class="code" >-<var class="Arg" >n</var > - 1</code > in the same block; see <a href="chapBib_mj.html#biBKudryavtseva2011aa" >[KM11]</a> and <a href="chapBib_mj.html#biBKudryavtseva2015aa" >[KMU15]</a>.</p>code class="func" >IsBlockBijection</code > (<a href="chap3_mj.html#X829494DF7FD6CFEC" ><span class="RefLink" >3.5-16</span ></a>).</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >Number(PartitionMonoid(3), IsBlockBijection);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >S := DualSymmetricInverseSemigroup(3);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Size(S);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >S := PartialDualSymmetricInverseMonoid(5);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Size(S);</span >pre ></div >"X8301C61384168D6F" name="X8301C61384168D6F" ></a></p>div class=<inverse blockbijectionmonoid degree4with3generators;div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣< block bijection of 4with generators>tr class"" ><code class="func" > SingularUniformBlockBijectionMonoid>( <ar classArg/> )</td < class="tdright" >( operationnbsp</d</r</></div >div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ PartialUniformBlockBijectionMonoiddiv classfunc< class"unc" width10%>tr td ="tdleft" >code class="func" ̻ SingularFactorisableDualSymmetricInverseMonoid> var ="Arg" n<var </d>td ="tdright> operationnbsp)/td> div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ PlanarUniformBlockBijectionMonoid</code >( <var class="Arg" >n</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" >‣ SingularPlanarUniformBlockBijectionMonoid</code >( <var class="Arg" >n</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >var class="Arg" >n</var > is a positive integer, then this operation returns the uniform block bijection monoid of degree <var class="Arg" >n</<panclass"APprompt" >gtspan span => SymmetricInverseMonoid(5);<span code class="code" >SingularUniformBlockBijectionMonoid"ode>SingularPlanarUniformBlockBijectionMonoid"n/ar least ./p> code class="code" >PartialUniformBlockBijectionMonoid</code > returns the submonoid of the uniform block bijection monoid of degree <code class="code" ><var class="Arg" >n</var > + 1</code > consisting of those uniform block bijection with <code class="code" ><var class="Arg" >n</var > + 1</code > and <code class="code" >-<var class="Arg" >n</var > - 1</code > in the same block.</p>"func" >IsUniformBlockBijectioncode > (< href="chap3_mjhtml#" ><panclassRefLink>.-7/><a)<pjava.lang.StringIndexOutOfBoundsException: Index 145 out of bounds for length 145span class="< class=""< =func" &82;SingularPlanarPartitionMonoid>(var =Arg><var )/d> classtdright(nbsp <td <tr <table <div java.lang.StringIndexOutOfBoundsException: Index 237 out of bounds for length 237span class="GAPprompt" >gap></span > <span class="GAPinput" >Size(PlanarUniformBlockBijectionMonoid(8));</span >span class="GAPprompt" >gapspan classspan class="GAPprompt" >gapdiv class="example" <pre >span class="GAPprompt" >gap></span > <span &;regular monoid degree 5generatorsgt;span class="GAPprompt" >gap></span > <span span =GAPprompt">&;span> =" GAPinput:()</>span span ="" >apgt;/span > <span class="GAPinput" >ize<span span class="GAPprompt" >gap></span > <span class<span class="GAPprompt" >gap><span span classGAPinputDifference(java.lang.StringIndexOutOfBoundsException: Range [75, 74) out of bounds for length 86span class="GAPprompt" >gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0span class="GAPprompt" >gap></span > <span class="GAPinput" >NrIdempotents(S);</span >span class="GAPprompt" >gap></span ><div class="" ><table ="" width="100" <>< classtdleft<odeclass=func&82;SingularModularPartitionMonoidcode > <ar="" m<var ,<arclass"Arg" ><var <td <td ="tdright> operationnbsp;) v>"X8444092A7967A029" name="X8444092A7967A029" ></a></p>div class="func" ><table classdiv class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ SingularPlanarPartitionMonoid</code >( <var class="Arg" >n</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >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 defined in Chapter <a href="chap3_mj.html#X7C18DB427C9C0917" ><span class="RefLink" >3</span ></a>).</p>code class="code" >SingularPlanarPartitionMonoid</code > returns the ideal of the planar partition monoid consisting of the non-invertible elements (i.e. those not in the group of units).</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >S := PlanarPartitionMonoid(3);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Size(S);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >T := SingularPlanarPartitionMonoid(3);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Size(T);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Difference(S, T);</span >pre ></div >"X7F208DC584C0B9D1" name="X7F208DC584C0B9D1" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ ModularPartitionMonoid</code >( <var class="Arg" >m</var >, <var class="Arg" >n</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >"func" >table ="" ="0%> var class="><var /d> classtdright(;operationnbsp/d<tr /></>div class="func" ><iv =C"java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25 div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ SingularPlanarModularPartitionMonoid</code >( <var class="Arg" >m</var >, <var class="Arg" >n</var > )</td span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7B1CD5FC7E034B88" >7.9-3 IsFreeBand</a></span >Messung V0.5 C=100 H=100 G=100
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