<!-- ------------------------------------------------------------------- --> <!-- --> <!-- intro.xml XModAlg documentation Z. Arvasi --> <!-- & A. Odabas --> <!-- Copyright (C) 2014-2021, Z. Arvasi & A. Odabas, --> <!-- Osmangazi University, Eskisehir, Turkey --> <!-- --> <!-- ------------------------------------------------------------------- -->
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<Chapter Label="Intro">
<Heading>Introduction</Heading>
In 1950 S. MacLane and J.H.C. Whitehead, <Cite Key="whitehead"/>
suggested that crossed modules modeled homotopy <M>2</M>-types.
Later crossed modules have been considered as
<M>2</M><E>-dimensional groups</E>, <Cite Key="brown1"/>,
<Cite Key="brown2"/>.
The commutative algebra version of this construction has been adapted by
T. Porter, <Cite Key="arvasi2"/>, <Cite Key="porter1"/>.
This algebraic version is called <E>combinatorial algebra theory</E>,
which contains potentially important new ideas
(see <Cite Key="shammu1"/>, <Cite Key="arvasi2"/>, <Cite Key="arvasi3"/>,
<Cite Key="arvasi4"/>).
<P/>
A share package <Package>XMod</Package>, <Cite Key="alp3"/>,
<Cite Key="alp2"/>, was prepared by
M. Alp and C.D. Wensley for the &GAP; computational group theory language,
initially for &GAP;3 then revised for &GAP;4.
The <M>2</M>-dimensional part of this programme contains functions for computing
crossed modules and cat<M>^{1}</M>-groups and their morphisms
<Cite Key="alp3"/>.
<P/>
This package includes functions for computing crossed modules of algebras,
cat<M>^{1}</M>-algebras and their
morphisms by analogy with <E>computational group theory</E>.
We will concentrate on group rings over of abelian groups over finite fields because these algebras are conveniently implemented in &GAP;.
The tools needed are the group algebras in which the group algebra functor
<M>\mathcal{K}(.):Gr\rightarrow Alg</M>
is left adjoint to the unit group functor
<M>\mathcal{U}(.):Alg\rightarrow Gr</M>.
<P/>
The categories <C>XModAlg</C> (crossed modules of algebras)
and <C>Cat1Alg</C> (cat<M>^{1}</M>-algebras) are equivalent,
and we include functions to convert objects and morphisms between them.
The algorithms implemented in this package are analyzed in
A. Odabas's Ph.D. thesis,
and described in detail in the paper <Cite Key="arvasi_odabas" />.
<P/>
There are aspects of commutative algebras for which no &GAP; functions yet exist, for example semidirect products.
We have included here functions for all homomorphisms of algebras.
</Chapter>
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