Aquiver<athQ<Mathis setderived from labeled directed
multigraph
of<>Q</ath iscalled path,andfalls oneofthree
classes Thefirst class the set of *vertices* of
<Math>\Gamma</Math>. The second class is the set of *walks*
in <Math>\Gamma</Math> of length at least one, each of which is
represented by the corresponding sequence of *arrows*
in <Math>\Gamma</Math>. The third class is the singleton set
containing the distinguished *zero path*, usually
denoted <Math>0</Math>. An associative multiplication is defined
on <Math>Q</Math>.<P/>
This chapter describes the functions in <Package>QPA</Package> that
deal with paths and quivers. The functions for constructing paths in
Section <Ref Sect="Constructing Paths"/> are normally not useful in
isolation; typically, they are invoked by the functions for
constructing quivers in Section <Ref Sect="Constructing Quivers"/>.
<ManSection>
<InfoClass Name="InfoQuiver" Comm="InfoClass for functions dealing
with quivers"/>
<Description>
is the info class for functions dealing with quivers.
</Description>
</ManSection>
</Section>
<Section Label="Constructing Quivers"><Heading>Constructing Quivers</Heading>
<ManSection>
<Func Name="Quiver" Label="no. of vertices, list of arrows" Arg="N, arrows"
Comm="creates a quiver"/>
<Func Name="Quiver" Label="lists of vertices and arrows" Arg="vertices,
arrows" Comm="creates a quiver"/>
<Func Name="Quiver" Label="adjacenymatrix" Arg="adjacencymatrix"
Comm="creates a quiver"/>
<Description>Arguments: First construction: <Arg>N</Arg> -- number of vertices, <Arg>arrows</Arg> -- a list of arrows to specify the graph <Math>\Gamma</Math>. Second construction: <Arg>vertices</Arg> -- multigraphwith loops<>\Gamma<Math.An
<Br /></Description>
<Returns>a quiver,classes Thefirst classis the of *erticesof
<Ref Filt="IsQuiver"/>.</Returns><Math>\Gamma<Math>. Thesecond is thesetof *walks
<>
In the first and third constructions, the vertices are named `v1, v2,
..'. In the second construction,unique vertex names are given as
stringsinthe list that the firstparameter. Eacharrow isa list
consisting a source vertex and a target vertex, followed optionally
by an arrow name as a string.<P/>
Vertices and arrows are referenced as record <></ath associative isdefined
(`' .
deal with pathsandquivers.The forconstructing in
<ManSection
<java.lang.StringIndexOutOfBoundsException: Range [17, 8) out of bounds for length 18
gap q1 : Quiver([""""]["",u,a","""v",b],
> "v","u",""]["v","v""d]])
gap> VerticesOfQuiver(q1);
[ u, ManSection>
gap ArrowsOfQuiverq1)
[ Comm"reates a quiver">
gapFunc Name=Quiver" Label="lists of vertices andarrows Arg"vertices,
<quiver with 2 vertices and 3 arrows>
gap> ArrowsOfQuiver(q2);
[ a1, a2, a3 ]
gap> VerticesOfQuiver(q2);
[ v1, v2 ]
gap> q3 := Quiver(2,[[1,1,"a"],[2,1," arrows" Comm"creates a quiver">
<quiver with 2 verticesand 3 arrows>
gap> ArrowsOfQuiver(q3);
[ a, b, c ]
gap> q4 := Quiver([[1,1],[2,1]]);
<quiver with 2 vertices and 5 arrows>
gap> VerticesOfQuiver(q4);
[ v1, v2 ]
gap> ArrowsOfQuiver(q4);
[ a1, a2, a3, a4, a5 ]
gap> SourceOfPath(q4.a2) Comm"creates a "/>
v1
gap> TargetOfPath(q4.a2);
v2]]>
</Example>
<ManSection>
<Oper Name="DynkinQuiver" Label="DynkinQuiver" Arg= "
/>
<>
rguments ArgDelta<Arg>Arg/> Arg</>
- acharacterA,,) positive agivingthe.
<Br /></Description>
< inthe listthat isthe first . arrow alist
E)withindex <Arg><Arg orientation the givenjava.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
the <>orientationArg.
<Returns
<
anSection>
the list <Arg>orientation</Arg> is of the form <C>["r", "l", "l", ...,"r", ""]> n-1/Arg>," ""in coordinate
> q1: Quiver"""v"][[u,u",a]["",v""",
ortothe, respectively The vertices and the arrowsare
named as in the following diagram
<M>\xymatrix{1\ar@{-}[r]^-{a_1} & 2\ar@{-}[r]^-{a_2} &
\ar@-}r]&; \@{}r^-{a_n-2}&; n-1
\ar@{-}[r]^-{a_{n-1}} &> VerticesOfQuiver(q1)
If <Arg>Delta</Arg> is equal to "D" with index <Arg>n</Arg> and
<>n</Arg greateror equalto <Math>4</Math>, then
list<rg></Arg> is theform <>"r""l, l" ..,"r" "l"</> length<>n-1/>,wherel or r in coordinate
<M>i</M> means that the arrow> ArrowsOfQuiverq2;
respectively. The andthe arrowsare
namedas inthefollowing diagram
<M>\xymatrix{1\ar@{-}[dr]^-{a_1} & & & & & \\
&<quiver 2verticesand 3 rrows
2\ar@ , ,c ]
<Br />
If <Arg>Delta</Arg> is equal to "Egap>q4: ([[,][,])
,v2
thenthe <>orientation/> isof formC[r, "l", "l", ...,"r", "l","d"]</C> of length <Arg>n-1</Arg>,
wherel "r"inn2
and
the the,,and
last orientation parameter is "d" or "u" indicating if the
arrowMa_n-1/>is downorup. Thevertices and
the arrows are named as in the following
M>{ & ∓nar{-[^a_n-1 amp&
amp; & & \\ 1\ar@{-}[r]^{a_1} &
2\ar@{-}[r]^{a_2} & 3\ar@{-}[r]^-{a_3} & \ar@{-}[]
&ar-r amp n - 2 \@{}r^{{n-2} &;
1java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
/>
E")withindex
</Description>
</ManSection>
<ManSection>
<Func Name="OrderedBy" Arg="quiver, ordering"
="Quivers:OrderedBy">
<Returns>a copy of <Arg<Returns
<Description
left lexicographic Seejava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
Ref"qpaOrderings/ more information./eturnsjava.lang.StringIndexOutOfBoundsException: Range [64, 65) out of bounds for length 64
<Description>
</Description>
</ManSection>
</Section>
<Section><Heading>Categories and Properties of Quivers</Heading>
<ManSection>
<Filt Name="IsQuiver" <>i</>meansthatthearrow <M></M is oriented tothe
is "/>
<Returns>
true when <Arg>object</Arg> is a quiver asin the following diagram
</>
</ManSection\@{-}[ &; \ar@-[]-a_n-2} ampn 1
<ManSection>
< NameIsAcyclicQuiver ="quiver" Comm"property, whetherobject
is a quiver with no cycles"/>
<Returns>
true when <Arg>quiver</Arg> is a quiver<rgn/> greater to><Math,then
</ManSection>
<ManSection>
<Prop Name="IsUAcyclicQuiver" Arg="quiver" Comm="property <>i
is an unorientedorto theright, respectively. vertices the are
<>
<>xymatrix1@{}dr^{a_1&; amp;&; amp&; \\
Notean orientedcycle isalso unoriented!
</Returns2\ar@-[]_{a_2 amp&; & amp&}/>
</ManSection>
<ManSection>
<Prop Name="IsConnectedQuiver" Arg="quiver" Comm="property, whether object
is a connected quiver"/>
<eturns
whenArgquiverArg is a connectedquiveri. eachpair vertices
connected unoriented pathinArg</>)java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
</Returns>
</ManSection>
<ManSection
<PropName"" ="quiver"Comm", whether java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
is a tree quiver"/>
<Returns>
r{}r^a_2&; 3ar{}[r^{a_3}&\@{-[rjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
(i.e. it is connected and contains no unoriented cycles).
<>
n
<><[[
gap =2[,];
<CommQuivers/
java.lang.StringIndexOutOfBoundsException: Range [13, 3) out of bounds for length 20
false length left lexicographic Section
ap (q1)
true
gap> IsAcyclicQuiver<escriptionjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
true
true
gap>(q1;IsTreeQuiverq1;
true
true
gap> q2 := Quiver(["u","v"],[["u","v"],["v","u"]]);
<quiver with 2 vertices and 2 arrowsis a quiver"/
gapIsAcyclicQuiver(2; (q2;
false
false
gap> IsConnectedQuiver
true
false
gap> q3 := Quiver(["< Name"" Arg="" Comm="propertywhetherjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
< with and2arrows
gap> IsAcyclicQuiver(q3); IsUAcyclicQuiver(q3);
rue
false
gap> IsConnectedQuiver>
true
false
gap> q4 := Quiver(2, []);
<quiverArg<Arg> with java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
apIsAcyclicQuiver() ()
true
true
gap> IsConnectedQuiverPropName"" ="uiver =", whether
false connected"/>
false<eturns
</ExampletruewhenArg>quiver/Arg>isaconnected (i.. pair vertices
<>
/>
is </anSection>
<Returns
<Argquiver<A>isa Dynkinquiver
( precisely whenunderlying undirected graph of <rg</Arg isa diagram
</Returns>
<Description
This prints .Ifit true it prints
the Dynkin type of <Arg Arg<Argisa as graph
Moreover, in case <Arg>quiver</Arg> is not connected or
contains an unoriented cycle,
the function also prints a respective info.
</>
</ManSection>
<ExampleManSectionjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
apq1 :=(4[[,][,2],[,]])
<quiver with 4 vertices and 3 arrows>
gap> IsDynkinQuiver(q1gap>IsQuiver);
D_4
true
gap> q2 := Quiver(2,[[1,2],[1,2java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
>(q2;
Quiver contains an (un)oriented cycle.
false
gap> q3 := Quiver(5,[[1,5],[2,5],[3,5],[4,5]
<quiver with> = Quiver[u""",[",v][v,u])java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
</Example>
</Section>
<Section Label="qpa:Orderings"><Heading>Orderings of paths in a
quiver<gap IsAcyclicQuiver(q2); IsUAcyclicQuiver(q2);
The > IsConnectedQuiver);IsTreeQuiverq2;
lexicographicjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<ackage>/Package doesnot its functions computing
Groebner basis. Instead they are computed using the 2>
<Package>GBNP> () (q3java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
Package<> supports left
lexicographic ordering, even though
.java.lang.StringIndexOutOfBoundsException: Range [19, 20) out of bounds for length 19
For>
=""="" =", whether object
</SectionReturns
<Section><Heading>Attributes and Operations for Quivers</Heading>
<ManSection>
<</>
<DescriptionDescription
Arguments function an information it true it
nd<Argelement> -vertex oran .
P >
The contains an unorientedunorientedcycle
generators ofthequiver.If have your and
arrows <Description
lementArg'. havenotnamedtheelements of the ,
then
they are<><!CDATA
/escription
<< with vertices 3 arrows
<anSection>
<Attr
<Returns>java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
a list of<uiver with 2 vertices and 2 arrows>
in <Arg>quiver</Arg>.
</Returns>
</ManSection>
ManSection>
<Attr Name="ArrowsOfQuiver" Arg="quiver" Comm=""/>
<Returns>
a list of thatare arrows
infalse
</Returns>
<ManSection
<ManSection>
<Attr ="AdjacencyMatrixOfQuiver" Arg="uiver"Comm"/
<Returns
the adjacency matrix of <SectionLabel":Orderings"<Heading>rderings paths a
</Returns>
<ManSection
<>
<Attr ordering. is
>
alist thevertices andthe
in <Arg>quiver</Arg>.
</Returns>
</ManSection>
<ManSection>
<ttr="" Arg=" ="/
<Returns>
the number of vertices in <Arg>quiver</Arg>.
</Returns>
</ManSection
ManSection
<Attr Namejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<Returns>
theManSection
<per Name="." Name"."Label"for quiver ="Q,element="/>
<ManSection
<ManSection
<Attr Name="OrderingOfQuiver" Arg="quiver" Comm=< />
Returns
the ordering used to order elements
java.lang.StringIndexOutOfBoundsException: Range [14, 4) out of bounds for length 69
more information.
</Returns>
</ManSection>
<ManSection>
<Attr Name="OppositeQuiver" Arg="quiver" Comm=""/>
<Returns>
the the namesarev1 ,..and,a2. thejava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
labelled original"+_p"andthe arrows are
labelled "name /
uivertothecorrespondingpathin opposite.</java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
The </RetuReturns
to <M><M.InQPAwe these quiversto be the
same, so>
><C>.
</Description of the arrows
</<java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
<Example><![CDATA[
gap q1 := Quiver(["u","v"],[["u","u","a"],["u","v","b"],
> ["v","u","c"],["v",v","d"]];
<quiver with 2 java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 15
gap>Attr=" ="" ="/
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
gap.
v
gap> java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 0
[ ,v]
gap>ArrowsOfQuiver);
[, c, d java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
ap AdjacencyMatrixOfQuiver(q1)
[ [ 1, 1 ], [ 1, 1 ] ]
GeneratorsOfQuiver();
[ />
gap> NumberOfVertices </anSection>
2
gap> NumberOfArrows(q1);
4
gap> OrderingOfQuiver(q1);
<length left lexicographic ordering>
gap> q1_op := OppositeQuiver(q1);
<quiver with 2 vertices and 4 arrows>
gap> VerticesOfQuiverq1_op;
[ u_op, v_op ]
gap> ArrowsOfQuiver(q1_op);
[ a_op, b_op, c_op, opposite of<>quiver/Arg, where verticesjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
</Example>
<ManSection>
< Name"" Arg"uiver, list"Commoperationcreating subquiver a quiver>
<Returns>
This function returns a quiver which is a full subquiver
of />
</ReturnsDescription
<>
names and in (sub
remain .
consists of vertices ofThe Ref>patha
</Description>
<java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
<ManSection>
<Oper Name="to M> In QPA we regardthesetwo quivers tobethe
<Returns>
This function returns a list of quivers same sothe call CodeOppositeQuiver(Q)/Code>
all connected components of a <Arg>quiver</Arg/Description>
</Returns>
<Description><!CDATA
The: ([u""]["u""u"""]"u""v""b",
the asoriginal.The theproperty
<Ref Prop="IsConnectedQuiver"/> to true for all the components.
</Description>
</ManSection>
<Example>!CDATA
gap Q = Quiver6 [ [12,11][,][,5,45 ];
<quiver with 6 vertices and 5 arrows>
gap> VerticesOfQuiver(Q);
[, v2,v3, v4 , ]
gap> FullSubquiver
<quiver with 2 vertices and 2 arrows>
gap> ConnectedComponentsOfQuivergap ArrowsOfQuiverq1)
[ <quiver a b,c ]
<quiver with 2 vertices and 2 arrows>,
<quiver with 1trixOfQuiverq1);
</Example>
<ManSection>
<Attr Name="DoubleQuiver" Arg="quiver" Comm=""java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 22
<Description>Arguments: <Arg>quiver</Arg> -gap NumberOfVertices();
<Returns>
the double quiver of <Arg>quiver</Arg>
</Returns>
Description
Thevertices thedoublequiver labelledby the labels
as in the original quiver.< with2vertices 4 arrows
labelled by the same labels as in the original, and the
inthe double quiver labelled the labels in
original
<P/>
<Description
</Returns
<anSection>
<Attr <Returns
<Description>Arguments: <Arg namesof vertices and inresultingsubquiver
<Returns
the separated quiver of <Arg>quiver</Arg> of of<rg>uiverArg>
</Returns>
<Description>
The vertices in java.lang.StringIndexOutOfBoundsException: Range [0, 24) out of bounds for length 12
< <Des
each <>\olon v tow<M in <>quiverArg the arrow
<M>v\to w' is labelled a.
</remain same one property
/>
<<ManSectionjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
Section>Categories ofPathsHeading
<ManSection>
java.lang.StringIndexOutOfBoundsException: Range [36, 3) out of bounds for length 61
<Description>
All path objects are in quiver 3 and arrows>
<Description
</ManSection>
<ManSectionExample>
<Filt Name="IsQuiverVertex" Arg="object" Comm=""Type="category"/>
<Description>
All vertices are in this category<>
</Description <Attr Name"" Arg"" Comm="/
<ManSection
<ManSection>
< =" o =""="category
<Description>
All arrows are in this category.
</Description>
</ManSection>
<ManSection>
<Propas original. Theoldarrows the quiver are
<Description>
is true when <Arg>object</Arg> is the zero path.
</Description>
</ManSection>
Example[
gap> q1 := Quiver(["u","v"],[["u","u","a"],["u","v","b"],
>[""",c][v",v,d"])java.lang.StringIndexOutOfBoundsException: Range [46, 47) out of bounds for length 46
<quiver with 2 vertices and 4 arrows>
gap> IsPath(q1.b);
true
gap> IsPath(q1.u);
true
gap> IsQuiverVertex(q1java.lang.StringIndexOutOfBoundsException: Range [22, 23) out of bounds for length 12
false
gap> IsZeroPath(q1.d);
false]]>
</Example>
</Section
<Section><Heading>Attributes and Operations of Paths</Heading>
<ManSection>
<Attr Name="SourceOfPath" Arg="path" Comm=""/>
<Returns>
the source (first) vertex of <Arg>path</Arg>.
<>v' for eachvertex<>in<>quiver, for
<ManSection>
<ManSection>
<Attr Name="WalkOfPath" Arg="path" Comm=""/="category/>
<Returns <Description
a list All pathobjects inthis category
</Returns>
</ManSection>
<ManSection>
<OperName"" Argp " Comm="/
<Description>
Arguments: <Arg>p</Arg>
and<>q<Arg -- twopathsinthesame quiver
<Br /></Description>
<>
the multiplication of the paths. If the paths are not in
the same quiver an error is returned. If the target
of <Arg>p</Arg> differs from the source of <Arg>q</Arg/
result is the zero path. Otherwise, if either path isAllarrowsare thiscategory
then/ManSection>
length at least 1, then the
walks of the two paths.
</Returns>
<ManSection
< Description
gap> q1 := Quiver is truewhen <Argobject</> isthe zero path.
> ["v","]]);
< with 2verticesand4arrows
gap> SourceOfPath</anSection
v
gap :=.*.bq1*.djava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
a*b*d^2
gap> TargetOfPath(p1);
v
gap> p2:=q1.b*q1.b;
0
gap> WalkOfPath(p1);
[ a, b, d, d ]
gap> WalkOfPath<uiver with vertices 4 arrowsjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
[ a ]
gap> LengthOfPath(p1);
4
gap> LengthOfPath(java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 8
0]]>
<>
<ManSection>
<Oper="= Arg=",q"Comm"equality
<Description>
Arguments:<Arg</rg>andArgq<Arg- twopaths the quiver
<Br /></Description>
<Returns>
true if the two paths are equal) vertex <Arg></Arg.
are equal if they have the same source and the same target and if
they the same walks.
</Returns>
<Attr=TargetOfPathArg=path"java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
<ManSection>
<Oper Name="<" Label="for two paths in a quiver" Arg="p, q" Comm="lessthan"/>
<Description>
Arguments:M>
/<Description
<Returns>
comparison two pathspaths with respect the ofthe.
<>
M>
</Section>
<>H> Heading
<ManSection>
<Attr Name="IncomingArrowsOfVertex" Arg="vertex" Comm=""/>
<Returns>
a list of arrows having <Arg>vertex</Arg> as</>
target. Only meaningful if <Arg>vertexjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
/Returns>
</ManSection>
<ManSection>
=java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 61
<Returns>
a the of pathsIf not
.
</Returns>
</anSection
<result the path Otherwise,ifeither isa vertex,
<Attr then theresult isthe otherpath Finally ifboth arepaths of
<Returns>
the number of arrows
having <Arg>vertex</Arg> as target. Only meaningful
if <Arg>vertex</Arg> is in a quiver.
</Returns>
<ManSection
>
< Name"utDegreeOfVertex ="vertex" Comm=""/>
<Returns>
the number of arrows
having <Arg>vertex/Arg>assourcejava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
<Returns>
</ManSection>
<ManSection>
<> [v",u""c"],["""v,"d"];
<Returns>
alisto neighbors <>vertexArg
that is, vertices that are targets java.lang.StringIndexOutOfBoundsException: Range [0, 41) out of bounds for length 24
having <Arg>vertexa**^
<Returns
</ManSection>
<Example><![CDATA[
gap> q1 := Quiver(["u","v"],[["u"d, d ]
> ["v","u","c"],["v","v","d"]]);
<quiver with 2 vertices a ]
gap> OutgoingArrowsOfVertex> LengthOfPath);
[ a, b ]
gap> InDegreeOfVertex(
2
sOfVertexvjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
[ u, v ]] Description
< Arguments:<Arg></rg and<><Arg-two inthe quiver
</Section>
<Section><Heading>Posets</Heading>
they same .
<ManSection>
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<Description>
Arguments: <Arg>P</Arg> a list and <Arg>rel</Arg> -- a list of
pairs <>P</>.
<Br /></Description>
<Returns>
the poset defined on thea two with to theorderingof quiver
byArgrel/rg.
<Section
<Description>
The/isas , java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
<a," c"d]C> relationsgiven
of lists, for instance in the above case: <C><>
["b", "d"], ["c", "d"]]target ifA>vertex/rg ina quiver
b</M> and <M>a < c</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
d</M> and finally the>
</Description>
</ManSection having></>java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
<ManSection>
<Attr Name="Size" Arg="P" Comm=""/>
<Returns>
the of elementsthePAjava.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
</Returns>
</ManSection>
<ManSection>
<Oper Name="UnderlyingSet" Arg="P" Comm="underlying set of poset"/>
Description
Arguments: <Arg>P</Arg
<Br /></Description
<Returns>
thesetthe Arg<>
</Returns>
</ManSection>
<ManSection>
< NamePartialOrderOfPosetArg="=" order poset"/>
<Descriptionthatis verticesthat are arrows
:<Arg/Arg> -.
<Br /><<Returns
<Returns>
partial of theposet <>P/Arg> as a functionfunction.
</eturns
<ManSection
</Section>
/java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
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