The aim of this package is to provide operations for finite groupoids.
A <E>groupoid</E> is constructed from a group and a set of objects.
In order to provide a sequence of categories, with increasing structure,
mimicing those for groups, we introduce in this chapter the notions of
<E>magma with objects</E>; <E>semigroup with objects</E>
and <E>monoid with objects</E>.
The next chapter introduces morphisms of these structures.
At a first reading of this manual, the user is advised to skip quickly
through these first two chapters, and then move on to groupoids
in Chapter <Ref Chap='chap-gpd' />.
<P/>
The definitions of the standard properties of groupoids
can be found in Philip Higgins' book ``Categories and Groupoids''
<Cite Key="HiGpd" /> (originally published in 1971, reprinted by TAC in 2005),
and in Ronnie Brown's book ``Topology'' ,
revised and reissued as ``Topology and Groupoids''
<Cite Key="BrTopGpd" />.
<Section Label="sec-mwo">
<Heading>Magmas with objects; arrows</Heading>
A <E>magma with objects</E> <M>M</M> consists of
a set of <E>objects</E> Ob<M>(M)</M>,
and a set of <E>arrows</E> Arr<M>(M)</M>
together with <E>tail</E> and <E>head</E> maps
<M>t,h :</M> Arr<M>(M) \to</M> Ob<M>(M)</M>,
and a <E>partial multiplication</E>
<M>* :</M> Arr<M>(M) \to </M> Arr<M>(M)</M>,
with <M>a*b</M> defined precisely when the head of <M>a</M>
coincides with the tail of <M>b</M>.
We write an arrow <M>a</M> with tail <M>u</M> and head <M>v</M>
as <M>(a : u \to v)</M>.
<P/>
When this multiplication is associative we obtain a
<E>semigroup with objects</E>.
<P/>
A <E>loop</E> is an arrow whose tail and head are the same object.
An <E>identity arrow</E> at object <M>u</M> is a loop <M>(1_u : u \to u)</M>
such that <M>a*1_u=a</M> and <M>1_u*b=b</M>
whenever <M>u</M> is the head of <M>a</M> and the tail of <M>b</M>.
When <M>M</M> is a semigroup with objects
and every object has an identity arrow,
we obtain a <E>monoid with objects</E>,
which is just the usual notion of mathematical category.
<P/>
An arrow <M>(a : u \to v)</M> in a monoid with objects
has <E>inverse</E> <M>(a^{-1} : v \to u)</M> provided
<M>a*a^{-1} = 1_u</M> and <M>a^{-1}*a = 1_v</M>.
A monoid with objects in which every arrow has an inverse
is a <E>group with objects</E>, usually called a <E>groupoid</E>.
<ManSection>
<Func Name="MagmaWithObjects"
Arg="args" />
<Oper Name="SinglePieceMagmaWithObjects"
Arg="magma, obs" />
<Attr Name="ObjectList"
Arg="mwo" Label="for magmas with objects" />
<Attr Name="RootObject"
Arg="mwo" Label="for magmas with objects" />
<Description>
The simplest construction for a magma with objects <M>M</M>
is to take a magma <M>m</M> and an ordered set <M>s</M>,
and form arrows <M>(u,a,v)</M> for every <M>a</M> in <M>m</M>
and <M>u,v</M> in <M>s</M>.
Multiplication is defined by <M>(u,a,v)*(v,b,w) = (u,a*b,w)</M>.
In this package we prefer to write <M>(u,a,v)</M> as <M>(a : u \to v)</M>,
so that the multiplication rule becomes
<M>(a : u \to v)*(b : v \to w) = (a*b : u \to w)</M>.
<P/>
Any finite, ordered set is in principle acceptable as
the object list of <M>M</M>, but most of the time we find it convenient
to restrict ourselves to sets of non-positive integers.
<P/>
This is the only construction implemented here for magmas, semigroups,
and monoids with objects, and these all have the property
<C>IsDirectProductWithCompleteDigraph</C>.
There are other constructions implemented for groupoids.
<P/>
The <E>root object</E> of <M>M</M> is the first element in <M>s</M>.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> tm := [[1,2,4,3],[1,2,4,3],[3,4,2,1],[3,4,2,1]];;
gap> Display( tm );
[ [ 1, 2, 4, 3 ],
[ 1, 2, 4, 3 ],
[ 3, 4, 2, 1 ],
[ 3, 4, 2, 1 ] ]
gap> m := MagmaByMultiplicationTable( tm );; SetName( m, "m" );
gap> m1 := MagmaElement(m,1);; m2 := MagmaElement(m,2);;
gap> m3 := MagmaElement(m,3);; m4 := MagmaElement(m,4);;
gap> M78 := MagmaWithObjects( m, [-8,-7] );
magma with objects :-
magma = m
objects = [ -8, -7 ]
gap> SetName( M78, "M78" );
gap> [ IsAssociative(M78), IsCommutative(M78) ];
[ false, false ]
gap> [ RootObject( M78 ), ObjectList( M78 ) ];
[ -8, [ -8, -7 ] ]
]]>
</Example>
<ManSection>
<Filt Name="IsDomainWithObjects"
Arg="obj"Type='Category'/>
<Filt Name="IsMagmaWithObjects"
Arg="obj"Type='Category'/>
<Description>
The output from function <C>MagmaWithObjects</C> lies in the categories
<C>IsDomainWithObjects</C>, <C>IsMagmaWithObjects</C> and
<C>CategoryCollections(IsMultiplicativeElementWithObjects)</C>.
As composition is only partial, the output does <E>not</E>
lie in the category <C>IsMagma</C>.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> [ IsDomainWithObjects(M78), IsMagmaWithObjects(M78), IsMagma(M78) ];
[ true, true, false ]
]]>
</Example>
<ManSection Label="man-arrow">
<Oper Name="Arrow"
Arg="mwo,elt,tail,head" />
<Oper Name="ElementOfArrow"
Arg="arr" Label="for magmas with objects" />
<Oper Name="TailOfArrow"
Arg="arr" Label="for magmas with objects" />
<Oper Name="HeadOfArrow"
Arg="arr" Label="for magmas with objects" />
<Description>
Arrows in a magma with objects lie in the category
<Index Key="IsMultiplicativeElementWithObjects">
<C>IsMultiplicativeElementWithObjects</C></Index>
<C>IsMultiplicativeElementWithObjects</C>.
An attempt to multiply two arrows which do not compose
resuts in <C>fail</C> being returned.
Each arrow <M>arr = (a : u \to v)</M> has three components.
The magma <E>element</E> <M>a \in m</M> may be accessed by
<C>ElementOfArrow(arr)</C>.
Similarly, the <E>tail</E> object <M>u</M> and the <E>head</E> object <M>v</M>
may be obtained using <C>TailOfArrow(arr)</C>
and <C>HeadOfArrow(arr)</C> respectively.
The operation <C>MultiplicativeElementWithObjects</C>
is a synonym for <C>Arrow</C>
since this was used in older versions of the package.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> a78 := Arrow( M78, m2, -7, -8 );
[m2 : -7 -> -8]
gap> a78 in M78;
true
gap> b87 := Arrow( M78, m4, -8, -7 );;
gap> [ ElementOfArrow( b87 ), TailOfArrow( b87 ), HeadOfArrow( b87 ) ];
[ m4, -8, -7 ]
gap> ba := b87*a78;; ab := a78*b87;; [ ba, ab ];
[ [m4 : -8 -> -8], [m3 : -7 -> -7] ]
gap> [ a78^2, ba^2, ba^3 ];
[ fail, [m1 : -8 -> -8], [m3 : -8 -> -8] ]
gap> ## this demonstrates non-associativity with a78*b87*a78:
gap> [ a78*ba, ab*a78, a78*ba=ab*a78 ];
[ [m3 : -7 -> -8], [m4 : -7 -> -8], false ]
]]>
</Example>
<ManSection>
<Prop Name="IsSinglePieceDomain"
Arg="mwo" />
<Prop Name="IsSinglePiece"
Arg="mwo" />
<Prop Name="IsDirectProductWithCompleteDigraph"
Arg="mwo" />
<Prop Name="IsDiscreteMagmaWithObjects"
Arg="mwo" />
<Description>
If the partial composition is forgotten, then what remains is a digraph
(usually with multiple edges and loops).
Thus the notion of <E>connected component</E> may be inherited
by magmas with objects from digraphs.
Unfortunately the terms <C>Component</C> and <C>Constituent</C>
are already in considerable use elsewhere in &GAP;,
so (and this may change if a more suitable word is suggested)
we use the term <C>IsSinglePieceDomain</C>
to describe an object with an underlying connected digraph.
The property <C>IsSinglePiece</C> is a synonym for
<C>IsSinglePieceDomain and IsMagmaWithObjects</C>.
When each connected component has a single object,
and there is more than one component,
the magma with objects is <E>discrete</E>.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> IsSinglePiece( M78 );
true
gap> IsDirectProductWithCompleteDigraph( M78 );
true
gap> IsDiscreteMagmaWithObjects( M78 );
false
]]>
</Example>
</Section>
<Section Label="sec-swo">
<Heading>Semigroups with objects</Heading>
A magma, semigroup, monoid, or group <M>M</M> can be made into
a magma with objects by the addition of a single object.
The two are algebraically isomorphic, and there is one arrow (a loop)
for each element in <M>M</M>. In the example we take the
transformation semigroup above, which has size <M>17</M>
at the object <M>0</M>.
<P/>
<P/>
<Example>
<![CDATA[
gap> S0 := MagmaWithSingleObject( sgp, 0 );
semigroup with objects :-
magma = sgp<t,s,r>
objects = [ 0 ]
gap> t0 := Arrow( S0, t, 0, 0 );
[Transformation( [ 1, 1, 2, 3 ] ) : 0 -> 0]
gap> Size( S0 );
17
]]>
</Example>
</Section>
<Section Label="monwo">
<Heading>Monoids with objects</Heading>
<ManSection>
<Func Name="MonoidWithObjects"
Arg="args" />
<Oper Name="SinglePieceMonoidWithObjects"
Arg="mon, obs" />
<Filt Name="IsMonoidWithObjects"
Arg="obj" />
<Description>
The constructions in section <Ref Sect="sec-mwo"/> give a
<C>SinglePieceMonoidWithObjects</C> when the magma is a monoid.
The example uses a finitely presented monoid with <M>2</M> generators
and <M>2</M> objects.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> fm := FreeMonoid( 2, "f" );;
gap> em := One( fm );;
gap> gm := GeneratorsOfMonoid( fm );;
gap> mon := fm/[ [gm[1]^3,em], [gm[1]*gm[2],gm[2]] ];;
gap> M49 := MonoidWithObjects( mon, [-9,-4] );
monoid with objects :-
magma = Monoid( [ f1, f2 ] )
objects = [ -9, -4 ]
gap> ktpo := KnownTruePropertiesOfObject( M49 );
[ "IsDuplicateFree", "IsAssociative", "IsSinglePieceDomain", "IsDirectProductWithCompleteDigraphDomain" ]
gap> catobj := CategoriesOfObject( M49 );;
[ "IsListOrCollection", "IsCollection", "IsExtLElement", "CategoryCollections(IsExtLElement)", "IsExtRElement", "CategoryCollections(IsExtRElement)", "CategoryCollections(IsMultiplicativeElement)", "IsGeneralizedDomain", "IsDomainWithObjects", "CategoryCollections(IsMultiplicativeElementWithObjects)", "CategoryCollections(IsMultiplicativeElementWithObjectsAndOnes)", "CategoryCollections(IsMultiplicativeElementWithObjectsAndInverses)", "IsMagmaWithObjects", "IsSemigroupWithObjects", "IsMonoidWithObjects" ]
]]>
</Example>
</Section>
<Section Label="genwo">
<Heading>Generators of magmas with objects</Heading>
<ManSection>
<Oper Name="GeneratorsOfMagmaWithObjects"
Arg="mwo" />
<Oper Name="GeneratorsOfSemigroupWithObjects"
Arg="swo" />
<Oper Name="GeneratorsOfMonoidWithObjects"
Arg="mwo" />
<Description>
For a magma or semigroup with objects, the generating set consists of arrows
<M>(g : u \to v)</M> for every pair of objects <M>u,v</M>
and every generating element for the magma or semigroup.
<P/>
For a monoid with objects, the generating set consists of two parts.
Firstly, there is a loop at the root object <M>r</M>
for each generator of the monoid.
Secondly, for each object <M>u</M> distinct from <M>r</M>,
there are arrows <M>(1 : r \to u)</M> and <M>(1 : u \to r)</M>.
(Perhaps only one of each pair is required?)
Then
<Display>
(e : u \to v ) = (1 : u \to r)*(e : r \to r)*(1 : r \to v).
</Display>
<P/>
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> GeneratorsOfMagmaWithObjects( M78 );
[ [m1 : -8 -> -8], [m2 : -8 -> -8], [m3 : -8 -> -8], [m4 : -8 -> -8],
[m1 : -8 -> -7], [m2 : -8 -> -7], [m3 : -8 -> -7], [m4 : -8 -> -7],
[m1 : -7 -> -8], [m2 : -7 -> -8], [m3 : -7 -> -8], [m4 : -7 -> -8],
[m1 : -7 -> -7], [m2 : -7 -> -7], [m3 : -7 -> -7], [m4 : -7 -> -7] ]
gap> genS := GeneratorsOfSemigroupWithObjects( S123 );;
gap> Length( genS );
27
gap> genM := GeneratorsOfMonoidWithObjects( M49 );
[ [f1 : -9 -> -9], [f2 : -9 -> -9], [<identity ...> : -9 -> -4],
[<identity ...> : -4 -> -9] ]
gap> g1:=genM[2];; g2:=genM[3];; g3:=genM[4];; g4:=genM[5];;
gap> [g4,g2,g1,g3];
[ [<identity ...> : -4 -> -9], [f2 : -9 -> -9], [f1 : -9 -> -9],
[<identity ...> : -9 -> -4] ]
gap> g4*g2*g1*g3;
[f2*f1 : -4 -> -4]
]]>
</Example>
</Section>
<Section Label="sec-pieces">
<Heading>Structures with more than one piece</Heading>
<ManSection>
<Oper Name="UnionOfPieces"
Arg="pieces" Label="for magmas with objects" />
<Attr Name="Pieces"
Arg="mwo" Label="for magmas with objects" />
<Oper Name="PieceOfObject"
Arg="mwo, obj" />
<Description>
A magma with objects whose underlying digraph has two or more connected
components can be constructed by taking the union of two or more
connected structures.
These, in turn, can be combined together.
The only requirement is that all the object lists should be disjoint.
The pieces are ordered by the order of their root objects.
<P/>
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> N1 := UnionOfPieces( [ M78, S123 ] );
magma with objects having 2 pieces :-
1: M78
2: semigroup with objects :-
magma = sgp<t,s,r>
objects = [ -3, -2, -1 ]
gap> ObjectList( N1 );
[ -8, -7, -3, -2, -1 ]
gap> Pieces(N1);
[ M78, semigroup with objects :-
magma = sgp<t,s,r>
objects = [ -3, -2, -1 ]
]
gap> PieceOfObject( N1, -7 );
M78
gap> N2 := UnionOfPieces( [ M49, S0 ] );
semigroup with objects having 2 pieces :-
1: monoid with objects :-
magma = Monoid( [ f1, f2 ] )
objects = [ -9, -4 ]
2: semigroup with objects :-
magma = sgp<t,s,r>
objects = [ 0 ]
gap> ObjectList( N2 );
[ -9, -4, 0 ]
gap> N3 := UnionOfPieces( [ N1, N2] );
magma with objects having 4 pieces :-
1: monoid with objects :-
magma = Monoid( [ f1, f2 ] )
objects = [ -9, -4 ]
2: M78
3: semigroup with objects :-
magma = sgp<t,s,r>
objects = [ -3, -2, -1 ]
4: semigroup with objects :-
magma = sgp<t,s,r>
objects = [ 0 ]
gap> ObjectList( N3 );
[ -9, -8, -7, -4, -3, -2, -1, 0 ]
gap> Length( GeneratorsOfMagmaWithObjects( N3 ) );
50
gap> ## the next command returns fail since the object sets are not disjoint:
gap> N4 := UnionOfPieces( [ S123, MagmaWithSingleObject( sgp, -2 ) ] );
fail
]]>
</Example>
</Section>
</Chapter>
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