A Math/> a**, into three. first isjava.lang.StringIndexOutOfBoundsException: Range [36, 34) out of bounds for length 53
with MathGamma<> Anelement
of <Math>Q</Math> is called a *path*, and falls into one of three
. first is set** java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
Math< class of*
in <Math>\Gamma</Math> of length at least Description
represented by the corresponding sequence of. ,are
in < the is . arrow list
containing the distinguished of sourcevertexjava.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
denotedMath>/>.Anassociative multiplication
on <Math>Q</Math>.<P/>
This chapter`)operator
deal with quivers functions paths
Section/>
isolation
Example><![CDATA[
<ManSection>
<InfoClass Name="InfoQuiver" Comm="InfoClass for functions dealing
with quivers"/>
<Description>
>q1=Quiver""v][u"""][u,v""",
</ [u",,v,";
</ManSection>
</Section>
<Section Label="Constructing Quivers"><
<anSection
<> (;
="aquiver/>
<= lists " ="java.lang.StringIndexOutOfBoundsException: Range [74, 75) out of bounds for length 74
=a/
<Func Name="Quiver" Label andjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
=aquiver
<Description>Arguments: java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
<Br /></Description>
<Returns>a quiver, orientation
<Ref Comm=""
Description
InArguments:<>Delta/Arg, <>n</rg,<>orientation/Arg
...'. In the second construction, unique vertex - (A,DE,a positive integer,and list orientation
ngs the that thefirstparameterEach is
consisting of a source vertex and "" index<Argn<>and ofthe arrows by
by an arrow name as a string.<P/>
Vertices and arrows are listArg</>
(`.< />
</<Description
java.lang.StringIndexOutOfBoundsException: Range [5, 2) out of bounds for length 15
<Example>< l"] of length"or "r in
gapq1 =(["u,][""",",u,v,b]
> ["v","u","c"],["v","v","d"]]); right. Theverticesandthe
<quiver with 2 verticesar{-[ ampar-[]-{{} amp
gapVerticesOfQuiverq1)
[ u, v ]
gap>
[ a, b,Arg> or to,
gap> q2 := Quiverthe ArgorientationArgisof form<[,""",.,",
<quiver with 2 vertices and 3"l]/>of <rg<Arg ""or""in coordinate
gap()
[ a1, a2, a3 ]
gap> VerticesOfQuiver. vertices the are
[ v1, v2 ]
gap asin followingjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
<quiverwith and3a>
gap> ArrowsOfQuiver(q3);
[a,b java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
=Quiver[1,21];
<quiver with 2 vertices and 5 arrows>
gap> VerticesOfQuiver(q4);
[v1 ]
gap> ArrowsOfQuiver( then the listArgorientation<Arg is of the <C>""
[ a1, a2, a3, a4, a5 ]
gap> SourceOfPath(q4where ""or" the n - 2/Arg> first coordinates
v1
gap> TargetOfPath(q4.a2 orientedtothe left orto the right respectively and the
v2]]>
</Example <>{}<M oriented up verticesand
<ManSection>
<>xymatrix;&mp \@{-[]{{}}&; amp
&mp-r
Comm &;\@{}[]&;ar-[]-a_n-2} ampn-
<Description>
Arguments 1}
-- a character (A,D,E), a positive integer, and a list giving the orientation.
<Br /></Description>
<Returns>a Dynkin quiver<M "" <><>and java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
the list <Arg>orientation</Arg CommQuivers/
<>
>
length. Section
the list <Arg>orientation</Arg> is of the form <C>["r< Sect=":">forinformation
java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
</ arrowa_i>isoriented the left
or to the right, respectively. The vertices and the arrows aquiver
named inthefollowingdiagram
<M>\xymatrix{1\ar@{-}[r]^-{<Returns
ar-}[]&ar@}r^{{}}&; n-1
\ar@{-}[r]^-{
If <ArgProp=""Argquiver=property java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
<>n<Arg or equal <Math4/Math> then
the list <Arg>orientation</Arg> is of the form <C "java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
<i<M>means Ma_i>is to java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
to the right,respectively. The and arrows
named Returns
<>{1ar-[]^{} amp&mp amp&; amp\\
&: an cycle alsoan cycle
2\ar@-ur-a_2}&; &&&; &<M>
<Br />
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 2
and <Arg>n</Arg> in <Math>[6,7,8]</Math>,
then the list <Arg>orientation<<> "truewhen <> (..each pair ofvertices
where is byanunoriented <>quiverArg)
and at coordinate <M>i</M> means that the arrow <M>a_i</M> is
oriented to the left or to the rightReturns>
<>
=IsTreeQuiverArg =propertyobject
the arrows are named as in the following diagram
<M>\xymatrix{ & & n\ar@{-}[d]^{a_{n-1}} & &
& & & \\ Returns
a@-[]{} amp3@{-[]-a_3 & ar-}r]
&\ar@{-}[r] & n - 2 \ar@{-}[r]^-{a_{n-2}} & n -
1}
</M>
<Br />
<Returns
</ManSection>>
<ManSectionExample!CDATA
<Funcgap> q1 :=Quiver(2[1,2])
Comm=":OrderedBy"/
<Returns>a("v1");
by <Arg>ordering</Arg>. The default ordering of a quiver is
lengthleft .See
<Ref Sect="qpa:Orderings"/g> IsQuiverq1)
<escription>
</Description>
</ManSection>
</Section>
<Section><Heading>IsConnectedQuiver) ();
<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<Filt Name="IsQuiver" Arg="object"Type="category" Comm="category, whether object
is aquiver/
<Returns>
true when <> IsAcyclicQuiver(2;IsUAcyclicQuiver)
</Returns>
</ManSection>
<ManSection>
Prop=IsAcyclicQuiverArgquiverComm, object
is a quiver with no cycles"/>
<Returns>
true when <Arg>quiver</quiver 2vertices 2>
</Returns>
</ManSection>
<ManSection>
true
is an java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 5
<Returns>
e when <Arg>quiver/isa quiverwith nounorientedcycles.
Note: an oriented cycle is also an unoriented cycle!
</Returns>
</ManSection>ap> IsAcyclicQuiver(q4) IsUAcyclicQuiverq4;
<ManSection>
<Prop =IsConnectedQuiverArg""Comm"property, object
a quiver
R>
<rg>quiver<Arg a quivere each of
is connectedManSection
<Returns
</anSection
<ManSection>quiver/rg a quiver
<Prop Name="IsTreeQuiver" Argmore, underlyingundirected graphof <>quiver> aDynkin).
is a tree quiver>
< This function anadditionalinformation If itreturns,
truewhen<>quiver/> a tree ajava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
(i.e. it is/escription
</Returns>
</ManSection>
<Example><![CDATA[
gap> q1 := Quiver(2,[[1,2]]);
<quiver with 2 vertices and 1 arrows>
gap> IsQuiver("v1")> q1 Quiver,14,42][4;
false
gap (q1java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
true
gap> IsAcyclicQuiver(q1); IsUAcyclicQuiver(q1);
true
true
gap> IsConnectedQuiver(q1);gap IsDynkinQuiver)
java.lang.StringIndexOutOfBoundsException: Range [6, 4) out of bounds for length 38
rue
gap q2:("",v]["""",""""];
<quiver with 2 vertices and 2 arrows>
> IsAcyclicQuiver();)
false
false
gap(q2 ();
true
false
gap> q3 := Quiver(["u","v"],[["u"<ackageQPA<Package>does have own for computing
<quiverwith2 verticesand arrowsjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
gapIsAcyclicQuiverq3;IsUAcyclicQuiver);
true
false
gap> IsConnectedQuiver(q3); IsTreeQuiver(q3);
true
false
gap> q4 := Quiver(2, []);provided bythe<Package>QPA/Package,onlysupports thelengthleft
<quivers>
gap> IsAcyclicQuiver(q4)moreorderings<P/>
true
true
gap> IsConnectedQuiver(q4); IsTreeQuiver(q4);
false
false]]>
</Example>
<ManSection
<PropName"sDynkinQuiver"ArgquiverCommpropertywhether
is a Dynkin quiver"/>
<>
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(more precisely, when underlying undirected graph of
<Returns
<>
This prints additional. If returns, itprints
the a ></Arg - a vertex anarrow
<P /
contains an unoriented ,
generators quiver you named vertices
/>
</e</>'. If you elements of quiver
Example[[
gap> q1 :=</>
<uiver 4 and >
gap> IsDynkinQuiver(q1);ManSection>
D_4
true
gap> q2 := Quiver(2,[[1,2],[1,2]]);
< 2andjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
gap>java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
Quiver contains an alistofpaths are
false
gap> q3 := Quiver/>
<quiver
</Example<AttrNameAdjacencyMatrixOfQuiver""="/
</Section>
< =qpa>Heading> of injava.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
/ManSection>
The only ManSection
lexicographicordering.The reasonforthisis that
<Package<Returns
Groebner a list ofthe verticesand arrows
<Package>GBNP</java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
provided < NameNumberOfVertices Arg=quiver Comm">
lexicographicReturnsjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
more orderings.<P/>
For constructing>
Explain how the vertices and <>
</Section>
<Section><Heading>Attributes and Operations for Quivers</Heading>
<>
O ="=for "ArgQ," Comm=">
<Description>
Arguments: <Arg>Q<//>
and <Arg>element>
< />
The operation <Ref Oper="." Label="for quiver"/> allows access <>
generatorsiver</Arg>. See Section <Ref Sect="qpa:Orderings"/> for
arrows then the access looksjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 21 element</Arg>'. If you have not named the elements of the quiver,
thenthe defaultnames are v1,v2 .. and a1 a2, ..in order
they are created.
</Description>
</ManSection>
<ManSection "nameinoriginal quiver +"_ thearrowsare
ANameVerticesOfQuiver"quiver"Comm"/>
<Returns>
a list of paths that are vertices
in <Arg>quiver</Arg>.
</eturns
/ManSection>
<ManSection>
<Attr Name="ArrowsOfQuiver" Arg="quiver" Comm=""/>
<Returns>
a list of paths , which the except every goes
inArgquiver<Arg>.
</Returns>
</ManSection>
<ManSection>
<Attr ="AdjacencyMatrixOfQuiver"ArgquiverComm">
q to the path the quiverP>
the adjacency matrix of <Arg>quiver</Arg>.
</Returns>
<to<MQ/> , regard two tobe the
<ManSection
<Attrreturns the object <CodeQ/ode.
<Returns>
a list verticesand the arrows
in <Arg>quiver</Arg>.
/Returns>
</ManSection>
<ManSection>
<Attr Name="NumberOfVertices" Arg="quiver" Comm=""/>
>
the number of vertices in v",d")
</Returns>
</ManSection>
<ManSection>
< Name"umberOfArrows"Argquiver Comm">
a
the number of arrows in <Arg>quiver</Arg>.
</Returns> q1v;
</ManSection>
<ManSection>
<Attr Name="u v ]
<Returns>(q1;
[ a, b]
in <Arg>quiver</Arg>. See Section <Ref Sectg>AdjacencyMatrixOfQuiverq1;
GeneratorsOfQuiverq1
</eturns
/anSection
<ManSection>
<Attr Name="OppositeQuiver" Arg="quiver" java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
<Returns ()java.lang.StringIndexOutOfBoundsException: Range [29, 30) out of bounds for length 29
the quiver <rg</Arg> the are
labelled "name in original quiver" + "_op" and the arrows are
labelled "name in original quiver" + Oper=FullSubquiverArg=" =" afull of "/java.lang.StringIndexOutOfBoundsException: Range [105, 106) out of bounds for length 105
<Returns
<>
This <escription
isThenames of verticesand arrows resulting(sub)quiver
the oppositedirection.P/>
The operation< Oper="OppositePath"/>takes a path in a
quiver to the corresponding path in the </ManSection>
The opposite of the opposite of a quiver <M>Q</M> is isomorphic
to<>/.In,we java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
, thecall<ode>(OppositeQuiver)<Code>
returns the object <Code>Q</Code>.
<Description
</ManSection>
<Example![
1 =Quiver""""][",u""][u""v,b]
> ["v","u","c remain the same in one. function sets the java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
<quiver with 2 verticesExample>![
gap> Q: (,,][,,32,45,4]])
a
v
v1 , ,v5v6]
java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 8
> (q1;
[a, ,d java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
(java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
[ [ 1, 1 ], [ 1, 1 ] ]
gap> GeneratorsOfQuiver(q1);
[ u, v, a, b, c, d ]
>NumberOfVerticesq1;
2
gap> NumberOfArrows(q1);
4
gap> OrderingOfQuiver(q1);
<length >
gap> in are by same
<uiver verticesand >
gap> VerticesOfQuiver(q1_op);
gap> ArrowsOfQuiver(q1_op);
[ a_op in thedoublequiverare by same as the
</Example>
<ManSection>
<Oper Name= />
<>
<anSection>
</>
<Description>
The ofvertices arrows ()
remain the same as in original one>
consists vertices <rgq</rg>
</Description>
</ManSection>
<ManSection>
<Oper Name="ConnectedComponentsOfQuiver" Arg="quiver" Comm="operation creating a list of all connected components of a quiver"/>
<Returns>
This function returns a list of quivers which are
all connected components of a <Arg>quiver</Arg>.
</Returns>
<Description>
The arrowMa\olon v\ w<> Arg</>java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
the asinoriginalone.The functionsets theproperty
<Ref Prop= </anSection
</Description>
/ManSection>
<Example><![CDATA[
gap> Q := Quiver(6, [ [1,2],[1,1],[3,2],[4,5],[4,5] ]);
<<><Heading andProperties </>
gap> VerticesOfQuiver(Q);
[ v1, v2, v3, v4, v5, v6 ]
gap> FullSubquiver(Q, [Q.v1, Q.v2]);Comm=""Type="category"/>
<quiver with 2 vertices and 2 arrows>
gap> ConnectedComponentsOfQuiver(Q);
[< with vertices 3arrows,
<quiver with 2 vertices and 2 arrows>, <>
<quiver
</Examplejava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
ManSection
<=DoubleQuiver=quiverComm">
<Description>Arguments: <Arg>quiver</Arg></>
<Returns>
the double quiver of <Arg>quiver<iltName"sArrow"Arg="bject"Comm""Typecategory"/>
</Returns>
<Description
The verticesManSection
as inthe quiver "" in double java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
labelled by the same labels
in<><![CDATAjava.lang.StringIndexOutOfBoundsException: Range [18, 19) out of bounds for length 18
original quiver with the string "bar" added.
> ",u""","",""");
</Description>
</ManSection>
<ManSection>
<Attr Name="SeparatedQuiver" Arg=
<Description>Arguments: <Arg>quiver<
<Returns>
the separated quiver of <Arg>quiver</Arg>.
</Returns>
<Description>
The vertices in the separated>
<M M>M> ArgArgand
each arrow <M>a\<ManSection
<M>v\java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</Description>
/java.lang.StringIndexOutOfBoundsException: Range [15, 16) out of bounds for length 15
</Section>
<Sectionthelength <rgpathArg>java.lang.StringIndexOutOfBoundsException: Range [34, 35) out of bounds for length 34
<ManSection>
pe"java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
>
All are this.
</Description>
</ManSection>
<ManSection>
<Filt Name="IsQuiverVertex" < =* =",q "/
<Description>
All vertices are in this category Arg<>-- same.
</<Returns
</ManSection>
<ManSection>
y">
<Description>
in .
</Description>
<ManSection
<ManSection>
<Prop Name="IsZeroPath" Arg="object" Comm="checks if />
<>
when>objectArg "u","c"],["v","v","d
</quiver2 and >
/>
<Example><![CDATAgap>p1=1aq1b*.dq1.d;
gap> q1 := Quiver(["u","v"],[["u","u","a"],["u","v","b"],
> ["v","u","c"],[java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
< 2 andarrows>
gap> IsPath(q1.b);
true
gap> IsPath(q1.u);
true
gap> IsQuiverVertex(q1.c);
false
gap> IsZeroPath(q1.d);
false]]>
</Example>
</Section>
<Section><Heading>Attributes and Operations of<Example
<ManSection< Name"Argp q"="/>
<Attr Name="SourceOfPath" Arg="path" Arguments >p/rg <>q/> - in same.
<Returns>
the source (first ofArgpath>java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
</Returns>
theyhavesamewalks
<ManSection>
<Attr Name"" Arg"path" Comm="/>
<Returns>
the target (last) vertex of <Arg
</Returns>
</ManSection>
<anSection
<Attr<Br//>
<Returns>
the a of the to ordering quiver
<Returns
</ManSection </anSection>
<ManSection>
<Attr Name="Section
<Returns>
a list of the arrows that constitute <Arg>path</Arg> in order.
<Returns
</ManSection>
<ManSection>
<>
<Description>
Arguments: <Arg>p</Arg>
and <Arg>q</Arg> -- two <AttrName"OutgoingArrowsOfVertex" Arg="vertex" Comm=""/>
<Br /></Description>
<Returns>
the multiplication the. If thepathsare in
the same quiver an error is returned. source
of <Arg>p</Arg> differs from the source </>
resultis zero. Otherwise path vertex
result the ., both pathsjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
length at least 1/>
walks of the
</ <ManSection
Attr="utDegreeOfVertex"Arg
<Example<> .
gap/java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
[,,c,[","d])java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
<quiver list f ofArg</>,
gap> SourceOfPath(q1.v);
v
gap> p1:=q1.a*q1.b*q1.d*q1.d;
abd^
gap> TargetOfPath(p1<>
v
gap> p2:=q1.b*q1.b;
0
gap> WalkOfPath(p1);
[ a, b, java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
gap> WalkOfPath(q1.a);
[a]
gap(p1
4
gap> LengthOfPath(q1.v);
]]>
</Example>
<ManSection(q1.);
<Oper Name="=" Arg="p, q" Comm="equality"/>
<>
Arguments Argp/>and <rgq/> - paths same.
<Br /></Description>
<Returns>
true if the two paths are equal. Two paths
are equal if
they havethesame walks
</Returns>
</ManSection>
<ManSection>
<Oper Name="<" Label="for two paths in a quiver" Arg="p, q" Comm="lessthan"/>
<Description>
Arguments: <Arg>p</Arg> and <Arg>q</Arg> fromArg/rg.
<Br /></Description>
<Returns>
comparisonofthe paths respectto the ordering the.
</Returns>
</ generated <Arg>rel</rg>.
</Section>
<Section><Heading>Attributes The elements in <Arg>P</rg> is given as alist andforexample
<ManSection>
<Attr <>["a","b"", ""and the relations are as alist
Returns
a list of arrows having <Arg>vertex</Arg> as
. Onlymeaningful if <rgvertex</> isin .
</Returns>
</ManSection>
<ManSection>
<Attr Name="OutgoingArrowsOfVertex" Arg="vertex" Comm=""/>
<Returnsjava.lang.StringIndexOutOfBoundsException: Range [12, 13) out of bounds for length 12
a listofarrows <Arg>vertex/rg as
source.
</Returns>
</ManSection>
<ManSection>
<Attr Name="InDegreeOfVertex" Arg="vertex" java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 12
<Returns>
the number of arrows
having <Arg>vertex</Arg> as number of poset <Arg>P</rg>.
if <Arg>vertex</Arg> is in a quiver.
</Returns>
</ManSection>
<ManSection>
<Attr Name="OutDegreeOfVertex" Arg="vertex" Comm=""/>
<Returns>
<>
having <Arg>vertex</Arg> as source.
</Returns>
</ManSection>
<ManSection>
<Attr Name="NeighborsOfVertex" Arg="vertex" Comm=" underlying of the poset P
<Returns
a Oper="" Arg"P"Commpartial of
, that targetsof arrows
having <Arg>vertex</Arg> as source Arguments: <Arg>P</--poset
<Returns>
</ManSection>
<Example><![the order posetArg<Arg
gap> q1/>
>/>
<quiver with 2 vertices and 4 arrows>
gap</hapter>
[ a, b ]
gap> InDegreeOfVertex(q1.u);
2
gap> NeighborsOfVertex(q1.v);
[ u, v ]]]>
</Example>
</Section>
<Section><Heading>Posets</Heading>
This implementation of posets was done by the HomAlg-project.
<ManSection>
<Oper Name="Poset" Label="for a list P and a set of relations rel" Arg="P, rel" Comm="poset"/>
<Description>
Arguments: <Arg>P</Arg> a list and <Arg>rel</Arg> -- a list of
pairs from <Arg>P</Arg>.
<Br /></Description>
<Returns>
the poset defined on the points <Arg>P</Arg> and the relations
generated by <Arg>rel</Arg>.
</Returns>
<Description>
The elements in <Arg>P</Arg> is given as a list, and for example
<C>["a", "b", "c", "d"]</C> and the relations are given as a list
of lists, for instance in the above case: <C>[ ["a", "b", "c"],
["b", "d"], ["c", "d"]].</C> The first list means that <M>a <
b</M> and <M>a < c</M>, and the second one means <M>b <
d</M> and finally the last one means <M>c < d</M>.
</Description>
</ManSection>
<ManSection>
<Attr Name="Size" Arg="P" Comm=""/>
<Returns>
the number of elements of the poset <Arg>P</Arg>.
</Returns>
</ManSection>
<ManSection>
<Oper Name="UnderlyingSet" Arg="P" Comm="underlying set of poset"/>
<Description>
Arguments: <Arg>P</Arg> -- poset.
<Br /></Description>
<Returns>
the underlying set of the poset <Arg>P</Arg>.
</Returns>
</ManSection>
<ManSection>
<Oper Name="PartialOrderOfPoset" Arg="P" Comm="partial order of poset"/>
<Description>
Arguments: <Arg>P</Arg> -- poset.
<Br /></Description>
<Returns>
the partial order of the poset <Arg>P</Arg> as a function.
</Returns>
</ManSection>
</Section>
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