<P/>
Given a pregroup <M>P</M> there is a universal group <M>\mathcal{U}(P)</M> that
contains <M>P</M>. The concept of a pregroup presentation is a generalisation of
presentations over the free group, that is a pregroup presentation is a way of
defining a group as a quotient of a universal group over a pregroup by giving
relator words over the pregroup.
<P/>
For the purposes of the RSym tester we introduce some more concepts.
<P/>
<Subsection Label="Chapter_Pregroup_Presentations_Section_Concepts_Subsection_Locations">
<Heading>Locations</Heading>
<P/>
A <E>location</E> on a pregroup relator <M> w = a_1a_2\ldots a_n</M> is an
index <M>i</M> between <M>1</M> and <M>n</M> and denotes the location between
<M>a_i</M> (the <Ref Attr="InLetter" Label="for IsPregroupLocation" />) and
<M>a_{i+1}</M> (the <Ref Attr="OutLetter" Label="for IsPregroupLocation" />), where
the relator is considered cyclically, that is, when <M>i=n</M> then the outletter
is <M>a_1</M>.
<P/>
</Subsection>
<P/>
A <E>place</E> <M>R(L, x, C)</M> on a pregroup relator <M>R</M> is a location
(<Ref Subsect="Chapter_Pregroup_Presentations_Section_Concepts_Subsection_Locations"/>) together with a letter from the pregroup and a
colour, which is either <E>red</E> or <E>green</E>.
</Subsection>
<P/>
<ManSection>
<Func Arg="pregroup, relators" Name="NewPregroupPresentation" />
<Returns>a pregroup presentation
</Returns>
<Description>
Creates a pregroup presentation over the <A>pregroup</A> with
relators <A>relators</A>.
</Description>
</ManSection>
<ManSection>
<Func Arg="F, rred, rgreen" Name="PregroupPresentationFromFp" />
<Returns>a pregroup presentation
</Returns>
<Description>
Creates a pregroup presentation over the pregroup defined by
<A>F</A> and <A>rred</A> with relators <A>rgreen</A>.
</Description>
</ManSection>
<ManSection>
<Func Arg="presentation" Name="PregroupPresentationToFpGroup" />
<Returns>a finitely presented group
</Returns>
<Description>
Converts the pregroup presentation <A>presentation</A> into
a finitely presented group.
</Description>
</ManSection>
<P/>
</Section>
<Section Label="Chapter_Pregroup_Presentations_Section_Filters_Attributes_and_Properties">
<Heading>Filters, Attributes, and Properties</Heading>
<ManSection>
<Filt Arg="arg" Name="" Label="for IsPregroupPresentation and IsComponentObjectRep and IsAttributeStoringRep"/>
<Returns><K>true</K> or <K>false</K>
</Returns>
<Description>
<P/>
</Description>
</ManSection>
</Section>
<Section Label="Chapter_Pregroup_Presentations_Section_Hyperbolicity_testing_for_pregroup_presentations">
<Heading>Hyperbolicity testing for pregroup presentations</Heading>
<P/>
<ManSection>
<Oper Arg="presentation, epsilon" Name="RSymTestOp" Label="for IsPregroupPresentation, IsRat"/>
<Description>
Test the group presented by <A>presentation</A> for hyperbolicity using
the RSym tester with parameter <A>epsilon</A>.
</Description>
</ManSection>
<ManSection>
<Func Arg="args..." Name="RSymTest" />
<Description>
<P/>
This is a wrapper for <Ref Oper="RSymTestOp" Label="for
IsPregroupPresentation, IsRat" />. If the first argument given is a free
group, the second and third lists of words over the free group, and the
fourth a rational, then this function creates a pregroup presentation from
the input data and invokes <Ref Oper="RSymTestOp" Label="for
IsPregroupPresentation, IsRat" /> on it. If the first
argument is a pregroup presentation and the second argument is rational
number, then it invokes <Ref Oper="RSymTestOp" Label="for
IsPregroupPresentation, IsRat" /> on that input.
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group2">
<Oper Arg="presentation" Name="IsHyperbolic" Label="for IsPregroupPresentation"/>
<Oper Arg="presentation, epsilon" Name="IsHyperbolic" Label="for IsPregroupPresentation, IsRat"/>
<Oper Arg="F, rred, rgreen, epsilon" Name="IsHyperbolic" Label="for IsFreeGroup, IsObject, IsObject, IsRat"/>
<Description>
Tests a given presentation for hyperbolicity using the RSym test procedure.
<P/>
</Description>
</ManSection>
</Section>
<Section Label="Chapter_Pregroup_Presentations_Section_Input_and_Output_of_Pregroup_Presentations">
<Heading>Input and Output of Pregroup Presentations</Heading>
<ManSection>
<Func Arg="presentation" Name="PregroupPresentationToKBMAG" />
<Returns>A KBMAG rewriting system
</Returns>
<Description>
Turns the pregroup presentation <A>presentation</A> into
valid input for Knuth-Bendix rewriting using KBMAG. Only
available if the kbmag package is available.
</Description>
</ManSection>
<ManSection>
<Func Arg="stream" Name="PregroupPresentationFromStream" />
<Returns>A pregroup presentation
</Returns>
<Description>
Reads a pregroup presentation from an input stream in the same format
that <Ref Func="PregroupPresentationToStream" /> uses.
<Example><![CDATA[
gap> stream := InputTextString(str);
InputTextString(0,146)
gap> PregroupPresentationFromStream(stream);
<pregroup presentation with 3 generators and 1 relators>
]]></Example>
</Description>
</ManSection>
<ManSection>
<Func Arg="stream, presentation" Name="PregroupPresentationToSimpleStream" />
<Description>
Writes the pregroup presentation <A>presentation</A> to
<A>stream</A>. Uses a simpler format than
<Ref Func="PregroupPresentationToStream" />
</Description>
</ManSection>
<ManSection>
<Func Arg="filename, presentation" Name="PregroupPresentationToFile" />
<Description>
Writes the pregroup presentation <A>presentation</A> to
file with name <A>filename</A>.
</Description>
</ManSection>
<ManSection>
<Func Arg="filename" Name="PregroupPresentationFromFile" />
<Description>
Reads a pregroup presentation from file with <A>filename</A>.
</Description>
</ManSection>
<ManSection>
<Func Arg="stream, presentation" Name="PregroupPresentationToSimpleFile" />
<Description>
Writes the pregroup presentation <A>presentation</A> to
file with name <A>filename</A> in a simple format.
</Description>
</ManSection>
</Section>
</Chapter>
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