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adjustments to the manual
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@cdwensley
cdwensley committed Jun 20, 2024
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122 changes: 60 additions & 62 deletions doc/cat1.xml
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Expand Up @@ -2,7 +2,7 @@
<!-- -->
<!-- cat1.xml XModAlg documentation Z. Arvasi -->
<!-- & A. Odabas -->
<!-- Copyright (C) 2014-2022, Z. Arvasi & A. Odabas, -->
<!-- Copyright (C) 2014-2024, Z. Arvasi & A. Odabas, -->
<!-- Osmangazi University, Eskisehir, Turkey -->
<!-- -->
<!-- ------------------------------------------------------------------- -->
Expand Down Expand Up @@ -37,9 +37,9 @@ algebra-algebroids, (Gaffar Musa's Ph.D. thesis, <Cite Key="mosa"/>).
In this section we describe an implementation of cat<M>^{1}</M>-algebras
and their morphisms.
<P/>
The notion of cat<M>^{1}</M>-groups was defined as an algebraic model
The notion of a cat<M>^{1}</M>-group was defined as an algebraic model
of <M>2</M>-types by Loday in <Cite Key="loday"/>.
Then Ellis defined the cat<M>^{1}</M>-algebras in <Cite Key="ellis1"/>.
Then Ellis defined cat<M>^{1}</M>-algebras in <Cite Key="ellis1"/>.
<P/>
Let <M>A</M> and <M>R</M> be <M>k</M>-algebras,
let <M>t,h:A\rightarrow R</M> be surjections,
Expand Down Expand Up @@ -72,7 +72,7 @@ The homomorphisms <M>t,h</M> and <M>e</M> are called the <E>tail map</E>,
Arg="t h" />
<Oper Name="PreCat1AlgebraByTailHeadEmbedding"
Arg="t h e" />
<Oper Name="PreCat1Algebra"
<Func Name="PreCat1Algebra"
Arg="args" />
<Prop Name="IsIdentityCat1Algebra"
Arg="C" />
Expand All @@ -81,14 +81,13 @@ The homomorphisms <M>t,h</M> and <M>e</M> are called the <E>tail map</E>,
<Prop Name="IsPreCat1Algebra"
Arg="C" />
<Description>
The operations listed above are used for construction of precat<M>^{1}</M>-
and cat<M>^{1}</M>-algebra structures.
The function <C>Cat1Algebra</C> selects the operation from the
above implementations up to user's input.
The operations <C>PreCat1AlgebraByEndomorphisms</C>
and <C>PreCat1AlgebraByTailHeadEmbedding</C>
are used with particular choices of algebra homomorphisms.

The operations listed above are used for the construction of
precat<M>^{1}</M>- and cat<M>^{1}</M>-algebras.
The functions <C>PreCat1Algebra</C> and <C>Cat1Algebra</C>
select the appropriate operation to suit the user's input.
</Description>
</ManSection>

Expand All @@ -98,14 +97,16 @@ are used with particular choices of algebra homomorphisms.
Arg="C" />
<Attr Name="Range" Label="for cat1-algebras"
Arg="C" />
<Attr Name="TailMap"
<Attr Name="TailMap" Label="for cat1-algebras"
Arg="C" />
<Attr Name="HeadMap"
<Attr Name="HeadMap" Label="for cat1-algebras"
Arg="C" />
<Attr Name="RangeEmbedding"
<Attr Name="RangeEmbedding" Label="for cat1-algebras"
Arg="C" />
<Meth Name="Kernel" Label="for cat1-algebras"
Arg="C" />
<Attr Name="KernelEmbedding" Label="for cat1-algebras"
Arg="C" />
<Attr Name="Boundary" Label="for cat1-algebras"
Arg="C" />
<Attr Name="Size2d" Label="for 2d-algebras"
Expand Down Expand Up @@ -161,35 +162,37 @@ Cat1-algebra [..=>..] :-

<ManSection>
<Oper Name="Cat1AlgebraSelect"
Arg="n gpsize gpnum num" />
Arg="GFnum gpsize gpnum num" />
<Description>
The <Ref Oper="Cat1Algebra"/> function may also be used to select a
cat<M>^{1}</M>-algebra from a data file.
All cat<M>^{1}</M>-structures on commutative algebras are stored in a list
in file <File>cat1algdata.g</File>.
The data is read into the list <C>CAT1ALG_LIST</C>
The <Ref Oper="Cat1Algebra"/> function may also be used to select certain
cat<M>^{1}</M>-group-algebras from the data file included with this package.
Data for these cat<M>^{1}</M>-structures on commutative group algebras is
stored in a list in file <File>cat1algdata.g</File>.
This data is read into the list <C>CAT1ALG_LIST</C>
only when this function is called.
<P/>
The function <C>Cat1AlgebraSelect</C> may be used in four ways:
<List>
<Item>
<C>Cat1AlgebraSelect( n )</C> returns the list of possible sizes of groups
for group algebras with Galois field <M>GF(n)</M>.
<C>Cat1AlgebraSelect( n )</C> returns the list of possible group orders when
Galois field <M>GF(n)</M>, with <M>n \in [2,3,4,5,7]</M>,
is used to form cat<M>^1</M>-group-algebra structures.
</Item>
<Item>
<C>Cat1AlgebraSelect( n, m )</C> returns the list of allowable
group numbers with given Galois field <M>GF(n)</M>
and groups of size <M>m</M>.
<C>Cat1AlgebraSelect( n, m )</C> returns the list of available
group numbers of size <M>m</M> with which to form
cat<M>^1</M>-group-algebra structures with given Galois field <M>GF(n)</M>.
</Item>
<Item>
<C>Cat1AlgebraSelect( n, m, k )</C> returns the list of possible
cat<M>^{1}</M>-algebra structures with given Galois field <M>GF(n)</M>
<C>Cat1AlgebraSelect( n, m, k )</C> prints the list of possible
cat<M>^{1}</M>-group-algebra structures with given Galois field <M>GF(n)</M>
and group number <M>k</M> of size <M>m</M>.
The number of these structures is returned.
</Item>
<Item>
<C>Cat1AlgebraSelect( n, m, k, j )</C>
(or simply <C>Cat1Algebra( n, m, k, j )</C>) returns the
<M>j</M>-th cat<M>^{1}</M>-algebra structure with these other parameters.
<M>j</M>-th cat<M>^{1}</M>-group-algebra structure with these other parameters.
</Item>
</List>
Now, we give examples of the use of this function.
Expand All @@ -200,31 +203,29 @@ Now, we give examples of the use of this function.
<Example>
<![CDATA[
gap> C := Cat1AlgebraSelect( 11 );
|--------------------------------------------------------|
| 11 is invalid number for Galois Field (GFnum) |
| Possible numbers for GFnum in the Data : |
|--------------------------------------------------------|
[ 2, 3, 4, 5, 7 ]
|--------------------------------------------------------|
| 11 is invalid value for the Galois Field (GFnum) |
| Available values for GFnum in the data : |
|--------------------------------------------------------|
[ 2, 3, 4, 5, 7 ]
Usage: Cat1Algebra( GFnum, gpsize, gpnum, num );
fail
gap> C := Cat1AlgebraSelect( 4, 12 );
|--------------------------------------------------------|
| 12 is invalid number for size of group (gpsize) |
| Possible numbers for the gpsize for GF(4) in the Data: |
|--------------------------------------------------------|
[ 1, 2, 3, 4, 5, 6, 7, 8, 9 ]
|--------------------------------------------------------|
| 12 is invalid value for size of group (gpsize) |
| Available values for gpsize with GF(4) in the data: |
|--------------------------------------------------------|
Usage: Cat1Algebra( GFnum, gpsize, gpnum, num );
fail
[ 1, 2, 3, 4, 5, 6, 7, 8, 9 ]
gap> C := Cat1AlgebraSelect( 2, 6, 3 );
|--------------------------------------------------------|
| 3 is invalid number for group of order 6 |
| Possible numbers for the gpnum in the Data : |
|--------------------------------------------------------|
[ 1, 2 ]
|--------------------------------------------------------|
| 3 is invalid value for groups of order 6 |
| Available values for gpnum for groups of size 6 : |
|--------------------------------------------------------|
Usage: Cat1Algebra( GFnum, gpsize, gpnum, num );
fail
[ 1, 2 ]
gap> C := Cat1AlgebraSelect( 2, 6, 2 );
There are 4 cat1-structures for the algebra GF(2)_c6.
There are 4 cat1-structures for the group algebra GF(2)_c6.
Range Alg Tail Head
|--------------------------------------------------------|
| GF(2)_c6 identity map identity map |
Expand All @@ -241,8 +242,9 @@ gap> C0 := Cat1AlgebraSelect( 4, 6, 2, 2 );
Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ] )]
gap> Size2d( C0 );
[ 4096, 1024 ]
gap> Dimension( C0 );
[ 6, 5 ]
gap> Display( C0 );
Cat1-algebra [GF(2^2)_c6=>..] :-
: source algebra has generators:
[ (Z(2)^0)*(), (Z(2)^0)*(1,2,3,4,5,6) ]
Expand Down Expand Up @@ -271,13 +273,13 @@ Cat1-algebra [GF(2^2)_c6=>..] :-

<ManSection>
<Oper Name="SubCat1Algebra"
Arg="arg" />
Arg="alg src rng" />
<Oper Name="SubPreCat1Algebra"
Arg="arg" />
<Prop Name="IsSubCat1Algebra"
Arg="arg" />
<Prop Name="IsSubPreCat1Algebra"
Arg="arg" />
Arg="alg src rng" />
<Oper Name="IsSubCat1Algebra"
Arg="alg sub" />
<Oper Name="IsSubPreCat1Algebra"
Arg="alg sub" />
<Description>
Let <M>\mathcal{C} = (e;t,h:A\rightarrow R)</M>
be a cat<M>^{1}</M>-algebra, and let <M>A^{\prime}</M>,
Expand Down Expand Up @@ -328,7 +330,6 @@ gap> CC3 := SubCat1Algebra( C3, AA3, BB3 );
[Algebra( GF(2), [ <zero> of ..., (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(4,5)
] ) -> Algebra( GF(2), [ <zero> of ..., (Z(2)^0)*() ] )]
gap> Display( CC3 );
Cat1-algebra [..=>..] :-
: source algebra has generators:
[ <zero> of ..., (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(4,5) ]
Expand Down Expand Up @@ -377,18 +378,20 @@ a cat<M>^{1}</M>-algebra morphism.


<ManSection>
<Oper Name="Cat1AlgebraMorphism"
<Func Name="Cat1AlgebraMorphism"
Arg="arg" />
<Func Name="PreCat1AlgebraMorphism"
Arg="arg" />
<Meth Name="IdentityMapping" Label="for cat1-algebras"
Arg="C" />
<Oper Name="PreCat1AlgebraMorphismByHoms"
Arg="f g" />
Arg="src rng srchom rnghom" />
<Oper Name="Cat1AlgebraMorphismByHoms"
Arg="f g" />
Arg="src rng srchom rnghom" />
<Prop Name="IsPreCat1AlgebraMorphism"
Arg="C" />
Arg="mor" />
<Prop Name="IsCat1AlgebraMorphism"
Arg="arg" />
Arg="mor" />
<Description>
These operations are used for constructing cat<M>^{1}</M>-algebra morphisms.
Details of the implementations can be found in <Cite Key="aodabas1"/>.
Expand Down Expand Up @@ -418,7 +421,6 @@ These are the six main attributes of a cat<M>^{1}</M>-algebra morphism.
gap> C1 := Cat1Algebra( 2, 1, 1, 1 );
[GF(2)_triv -> GF(2)_triv]
gap> Display( C1 );
Cat1-algebra [GF(2)_triv=>GF(2)_triv] :-
: source algebra has generators:
[ (Z(2)^0)*(), (Z(2)^0)*() ]
Expand All @@ -435,7 +437,6 @@ Cat1-algebra [GF(2)_triv=>GF(2)_triv] :-
gap> C2 := Cat1Algebra( 2, 2, 1, 2 );
[GF(2)_c2 -> GF(2)_triv]
gap> Display( C2 );
Cat1-algebra [GF(2)_c2=>GF(2)_triv] :-
: source algebra has generators:
[ (Z(2)^0)*(), (Z(2)^0)*(1,2) ]
Expand All @@ -453,7 +454,6 @@ Cat1-algebra [GF(2)_c2=>GF(2)_triv] :-
[ <zero> of ... ]
: kernel embedding maps generators of kernel to:
[ (Z(2)^0)*()+(Z(2)^0)*(1,2) ]
gap> C1 = C2;
false
gap> R1 := Source( C1 );;
Expand Down Expand Up @@ -514,7 +514,6 @@ this operation returns the image crossed module.
<![CDATA[
gap> imm := ImagesSource2DimensionalMapping( m );;
gap> Display( imm );
Cat1-algebra [..=>..] :-
: source algebra has generators:
[ (Z(2)^0)*(), (Z(2)^0)*() ]
Expand All @@ -527,7 +526,6 @@ Cat1-algebra [..=>..] :-
: range embedding maps range generators to:
[ (Z(2)^0)*() ]
: the kernel is trivial.
]]>
</Example>

Expand Down
38 changes: 19 additions & 19 deletions doc/xmod.xml
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Expand Up @@ -317,9 +317,9 @@ gap> Print( KnownAttributesOfObject(XIAk4), "\n" );

<ManSection>
<Oper Name="SubXModAlgebra"
Arg="X0" />
Arg="alg src rng" />
<Oper Name="IsSubXModAlgebra"
Arg="X0" />
Arg="alg sub" />
<Description>
A crossed module <M>\mathcal{X}^{\prime }
= (\partial ^{\prime }:S^{\prime}\rightarrow R^{\prime })</M>
Expand Down Expand Up @@ -407,26 +407,26 @@ by the function <C>Make2dAlgebraMorphism</C>.
<Func Name="XModAlgebraMorphism"
Arg="arg" />
<Meth Name="IdentityMapping" Label="for crossed modules of algebras"
Arg="X0" />
Arg="Xalg" />
<Oper Name="PreXModAlgebraMorphismByHoms"
Arg="f,g" />
Arg="src rng srchom rnghom" />
<Oper Name="XModAlgebraMorphismByHoms"
Arg="f,g" />
Arg="src rng srchom rnghom" />
<Prop Name="IsPreXModAlgebraMorphism"
Arg="f" />
Arg="mor" />
<Prop Name="IsXModAlgebraMorphism"
Arg="f" />
Arg="mor" />
<Attr Name="Source" Label="for morphisms of crossed modules of algebras"
Arg="m" />
Arg="mor" />
<Attr Name="Range" Label="for morphisms of crossed modules of algebras"
Arg="m" />
Arg="mor" />
<Meth Name="IsTotal" Label="for morphisms of crossed modules of algebras"
Arg="m" />
Arg="mor" />
<Meth Name="IsSingleValued"
Label="for morphisms of crossed modules of algebras"
Arg="m" />
Arg="m0r" />
<Meth Name="Name" Label="for morphisms of crossed modules of algebras"
Arg="m" />
Arg="mor" />

<Description>
These operations construct crossed module homomorphisms,
Expand Down Expand Up @@ -484,7 +484,7 @@ true

<ManSection>
<Meth Name="Kernel" Label="for morphisms of crossed modules of algebras"
Arg="X0" />
Arg="mor" />
<Description>
Let <M>(\theta,\varphi) : \mathcal{X} = (\partial : S \rightarrow R)
\rightarrow \mathcal{X}^{\prime} = (\partial^{\prime}
Expand Down Expand Up @@ -515,7 +515,7 @@ true

<ManSection>
<Oper Name="Image"
Arg="X0" />
Arg="mor" />
<Description>
Let <M>(\theta,\varphi) : \mathcal{X} = (\partial : S \rightarrow R)
\rightarrow \mathcal{X}^{\prime} = (\partial^{\prime} : S^{\prime}
Expand All @@ -541,15 +541,15 @@ induced from a given crossed module homomorphism.

<ManSection>
<Attr Name="SourceHom"
Arg="m" />
Arg="mor" />
<Attr Name="RangeHom"
Arg="m" />
Arg="mor" />
<Prop Name="IsInjective"
Arg="m" />
Arg="mor" />
<Prop Name="IsSurjective"
Arg="m" />
Arg="mor" />
<Prop Name="IsBijjective"
Arg="m" />
Arg="mor" />
<Description>
Let <M>(\theta,\varphi)</M> be a homomorphism of crossed modules.
If the homomorphisms <M>\theta</M> and <M>\varphi</M>
Expand Down
2 changes: 1 addition & 1 deletion lib/alg2obj.gd
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Expand Up @@ -2,7 +2,7 @@
##
#W alg2obj.gd The XMODALG package Zekeriya Arvasi
#W & Alper Odabas
#Y Copyright (C) 2014-2021, Zekeriya Arvasi & Alper Odabas,
#Y Copyright (C) 2014-2024, Zekeriya Arvasi & Alper Odabas,
##

############################## 2d-algebras ###########################
Expand Down
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