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ChrisJefferson merged 1 commit into from
Mar 2, 2018
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Adding examples to the relations chapter #2219

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72 changes: 72 additions & 0 deletions lib/relation.gd
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Original file line number Diff line number Diff line change
Expand Up @@ -95,6 +95,10 @@ DeclareSynonym("IsBinaryRelation",IsEndoGeneralMapping);
## to <A>list</A><C>[1]</C>, <M>2</M> to <A>list</A><C>[2]</C> and so on.
## The first version checks whether the list supplied is valid.
## The the <C>NC</C> version skips this check.
## <Example><![CDATA[
## gap> R:=BinaryRelationOnPoints([[1,2],[2],[3]]);
## Binary Relation on 3 points
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
Expand Down Expand Up @@ -137,6 +141,14 @@ DeclareGlobalFunction("RandomBinaryRelationOnPoints");
## <M>\{ 1, \ldots, <A>degree</A> \}</M>.
## In the second form, the objects <M>x</M> are from the domain
## <A>domain</A>.
## <Example><![CDATA[
## gap> IdentityBinaryRelation(5);
## <equivalence relation on <object> >
## gap> s4:=SymmetricGroup(4);
## Sym( [ 1 .. 4 ] )
## gap> IdentityBinaryRelation(s4);
## IdentityMapping( Sym( [ 1 .. 4 ] ) )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
Expand All @@ -156,6 +168,10 @@ DeclareGlobalFunction("IdentityBinaryRelation");
## consisting of the tuples collection <A>elms</A>.
## This construction is similar to <Ref Func="GeneralMappingByElements"/>
## where the source and range are the same set.
## <Example><![CDATA[
## gap> r:=BinaryRelationByElements(Domain([1..3]),[Tuple([1,2]),Tuple([1,3])]);
## <general mapping: <object> -> <object> >
## ]]></Example>
## </Description>
## </ManSection>
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@fingolfin fingolfin Feb 28, 2018

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How about also adding an example for EmptyBinaryRelation, e.g.

gap> EmptyBinaryRelation(3) = BinaryRelationOnPoints([ [], [], [] ]);
true

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@grouptheoryenthusiast grouptheoryenthusiast Feb 28, 2018

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Hi @fingolfin , thank you for the suggestion. I have now added this and also added Is Symmetric/Reflexive/Transitive BinaryRelation based on your comment above.
I've pushed these changes to this branch (as well as fixing a missing ;).

## <#/GAPDoc>
Expand All @@ -178,6 +194,10 @@ DeclareGlobalFunction("BinaryRelationByElements");
## the domain is the set of points <M>\{ 1, \ldots, <A>degree</A> \}</M>.
## In the second form, the domain is that given by the argument
## <A>domain</A>.
## <Example><![CDATA[
## gap> EmptyBinaryRelation(3) = BinaryRelationOnPoints([ [], [], [] ]);
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
Expand Down Expand Up @@ -214,6 +234,15 @@ DeclareGlobalFunction("EmptyBinaryRelation");
## In the last form, an isomorphic relation on points is constructed
## where the points are indices of the elements of the underlying domain
## in sorted order.
## <Example><![CDATA[
## gap> t:=Transformation([2,3,1]);;
## gap> r1:=AsBinaryRelationOnPoints(t);
## Binary Relation on 3 points
## gap> r2:=AsBinaryRelationOnPoints((1,2,3));
## Binary Relation on 3 points
## gap> r1=r2;
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
Expand Down Expand Up @@ -241,6 +270,11 @@ DeclareGlobalFunction("AsBinaryRelationOnPoints");
## <P/>
## The <Ref Func="Successors"/> of a relation is the adjacency list
## representation of the relation.
## <Example><![CDATA[
## gap> r1:=BinaryRelationOnPoints([[2],[3],[1]]);;
## gap> Successors(r1);
## [ [ 2 ], [ 3 ], [ 1 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
Expand All @@ -259,6 +293,10 @@ DeclareAttribute("Successors", IsBinaryRelation);
## returns the size of the underlying domain of the binary relation
## <A>R</A>.
## This is most natural when working with a binary relation on points.
## <Example><![CDATA[
## gap> DegreeOfBinaryRelation(r1);
## 3
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
Expand Down Expand Up @@ -306,6 +344,12 @@ DeclareSynonym("UnderlyingDomainOfBinaryRelation",Source);
## <P/>
## A reflexive binary relation is necessarily a total endomorphic
## mapping (tested via <Ref Func="IsTotal"/>).
## <Example><![CDATA[
## gap> IsReflexiveBinaryRelation(BinaryRelationOnPoints([[1,3],[2],[3]]));
## true
## gap> IsReflexiveBinaryRelation(BinaryRelationOnPoints([[2],[2]]));
## false
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
Expand All @@ -330,6 +374,12 @@ DeclareProperty("IsReflexiveBinaryRelation", IsBinaryRelation);
## Alternatively, <M>R</M> as a mapping is symmetric
## if for all <M>x \in X</M>, the preimage set of <M>x</M> under <M>R</M>
## equals the image set <M>R(x)</M>.
## <Example><![CDATA[
## gap> IsSymmetricBinaryRelation(BinaryRelationOnPoints([[2],[1]]));
## true
## gap> IsSymmetricBinaryRelation(BinaryRelationOnPoints([[2],[2]]));
## false
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
Expand All @@ -355,6 +405,12 @@ DeclareProperty("IsSymmetricBinaryRelation", IsBinaryRelation);
## Alternatively, <M>R</M> as a mapping is transitive if for all
## <M>x \in X</M>, the image set <M>R(R(x))</M> of the image
## set <M>R(x)</M> of <M>x</M> is a subset of <M>R(x)</M>.
## <Example><![CDATA[
## gap> IsTransitiveBinaryRelation(BinaryRelationOnPoints([[1,2,3],[2,3],[]]));
## true
## gap> IsTransitiveBinaryRelation(BinaryRelationOnPoints([[1,2],[2,3],[]]));
## false
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
Expand Down Expand Up @@ -735,6 +791,12 @@ DeclareAttribute("GeneratorsOfEquivalenceRelationPartition",
## and singletons will be ignored. The <C>NC</C> version will not check
## to see if the lists are pairwise mutually exclusive or that
## they contain only elements of the domain.
## <Example><![CDATA[
## gap> er:=EquivalenceRelationByPartition(Domain([1..10]),[[1,3,5,7,9],[2,4,6,8,10]]);
## <equivalence relation on <object> >
## gap> IsEquivalenceRelation(er);
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
Expand Down Expand Up @@ -867,6 +929,12 @@ DeclareAttribute("EquivalenceClassRelation", IsEquivalenceClass);
## <M>c1</M> and <M>c2</M> we may have <M>c1 = c2</M> without having
## <C>EquivalenceClasses</C><M>( c1 ) =
## </M><C>EquivalenceClasses</C><M>( c2 )</M>.
## <Example><![CDATA[
## gap> er:=EquivalenceRelationByPartition(Domain([1..10]),[[1,3,5,7,9],[2,4,6,8,10]]);
## <equivalence relation on <object> >
## gap> classes := EquivalenceClasses(er);
## [ {1}, {2} ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
Expand All @@ -889,6 +957,10 @@ DeclareAttribute("EquivalenceClasses", IsEquivalenceRelation);
## where <A>elt</A> is an element (i.e. a pair) of the domain of <A>rel</A>.
## In the <C>NC</C> form, it is not checked that <A>elt</A> is in the domain
## over which <A>rel</A> is defined.
## <Example><![CDATA[
## gap> EquivalenceClassOfElement(er,3);
## {3}
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
Expand Down