Quelle example21.xml Sprache: XML

gap> ######################### BEGIN COPYRIGHT MESSAGE #########################
GBNP - computing Gröbner bases of noncommutative polynomials
Copyright 2001-2010 by Arjeh M. Cohen, Dié A.H. Gijsbers, Jan Willem
Knopper, Chris Krook. Address: Discrete Algebra and Geometry (DAM) group
at the Department of Mathematics and Computer Science of Eindhoven
University of Technology.

For acknowledgements see the manual. The manual can be found in several
formats in the doc subdirectory of the GBNP distribution. The
acknowledgements formatted as text can be found in the file chap0.txt.

GBNP is free software; you can redistribute it and/or modify it under
the terms of the Lesser GNU General Public License as published by the
Free Software Foundation (FSF); either version 2.1 of the License, or
(at your option) any later version. For details, see the file 'LGPL' in
the doc subdirectory of the GBNP distribution or see the FSF's own site:
https://www.gnu.org/licenses/lgpl.html
gap> ########################## END COPYRIGHT MESSAGE ##########################

gap> ### filename = "example21.g"
gap> ### author Knoppe21

gap> ### THIS IS A GAP PACKAGE FOR COMPUTING NON-COMMUTATIVE GROBNER BASES

<#GAPDoc Label="Example21">
<Section Label="Example21"><Heading>The symmetric inverse monoid for a set of size four</Heading>
The algebra we will consider is from Example 4 from Linton <Cite Key="MR94k:20022"/>.
Its monomials form the symmetric inverse monoid, that is, the monoid of all
partial bijections on a given set, for a set of size four. The presentation
is <M>\langle s_1,s_2,s_3,e\mid s_i^2=(s_1s_2)^3=(s_2s_3)^3=(s_1s_3)^2=1, e^2=e,
s_1e=es_1,s_2e=es_2,es_3e=(es_3)^2=(s_3e)^2\rangle</M>. The dimension of the
monoid algebra is 209. The monoid has a natural representation of degree 4 by means of
partial permutation matrices, which can be obtained by taking
prefix relations <M>\{e,s_1-1, s_2-1, s_3e-s_3\}</M>.
<P/>
First load the package and set the standard infolevel <Ref
InfoClass="InfoGBNP" Style="Text"/> to 1 and the time infolevel <Ref
Func="InfoGBNPTime" Style="Text"/> to 1 (for more information about the info
level, see Chapter <Ref Chap="Info"/>).

<Listing><![CDATA[
gap> LoadPackage("gbnp", false);
true
gap> SetInfoLevel(InfoGBNP,1);
gap> SetInfoLevel(InfoGBNPTime,1);
]]></Listing>

Now enter the relations as GAP polynomials. The module is one dimensional so
it is possible to enter it with and without a module. In Example 18 (<Ref
Sect="Example18"/>) both ways are shown. Here the relations will be entered
without a module, with a free associative algebra with one over the field
(the rational numbers) (see also <Ref BookName="Reference"
Label="FreeAssociativeAlgebraWithOne"/>). For convenience we use the
variables <C>s1</C>, <C>s2</C>, <C>s3</C>, and <C>e</C> for the generators of
the algebra, and <C>o</C> for the identity element of the algebra.

<Listing><![CDATA[
gap> A:=FreeAssociativeAlgebraWithOne(Rationals, "s1", "s2", "s3", "e");
<algebra-with-one over Rationals, with 4 generators>
gap> g:=GeneratorsOfAlgebra(A);;
gap> s1:=g[2];;s2:=g[3];;s3:=g[4];;e:=g[5];;o:=g[1];;
]]></Listing>

It is possible to print symbols like they are printed in the algebra <C>A</C>
with the function <Ref Func="GBNP.ConfigPrint" Style="Text"/>:

<Listing><![CDATA[
gap> GBNP.ConfigPrint(A);
]]></Listing>

Now the relations are entered:

<Listing><![CDATA[
gap> twosidrels:=[s1^2-o,s2^2-o,s3^2-o,(s1*s2)^3-o,(s2*s3)^3-o,(s1*s3)^2-o,
>   e^2-e,s1*e-e*s1,s2*e-e*s2,e*s3*e-(e*s3)^2,e*s3*e-(s3*e)^2];
[ (-1)*<identity ...>+(1)*s1^2, (-1)*<identity ...>+(1)*s2^2,
  (-1)*<identity ...>+(1)*s3^2, (-1)*<identity ...>+(1)*(s1*s2)^3,
  (-1)*<identity ...>+(1)*(s2*s3)^3, (-1)*<identity ...>+(1)*(s1*s3)^2,
  (-1)*e+(1)*e^2, (1)*s1*e+(-1)*e*s1, (1)*s2*e+(-1)*e*s2,
  (1)*e*s3*e+(-1)*(e*s3)^2, (1)*e*s3*e+(-1)*(s3*e)^2 ]
gap> prefixrels:=[e,s1-o,s2-o,s3*e-s3];
[ (1)*e, (-1)*<identity ...>+(1)*s1, (-1)*<identity ...>+(1)*s2,
  (-1)*s3+(1)*s3*e ]
]]></Listing>

First the relations are converted into NP format (see <Ref Sect="NP"/>) and
next the function <Ref Func="SGrobnerModule" Style="Text"/> is called to
calculate a Gröbner basis record.

<Listing><![CDATA[
gap> GBR:=SGrobnerModule(GP2NPList(prefixrels),GP2NPList(twosidrels));;
#I  number of entered polynomials is 11
#I  number of polynomials after reduction is 11
#I  End of phase I
#I  End of phase II
#I  End of phase III
#I  End of phase IV
#I  The computation took 24 msecs.
#I  number of entered polynomials is 42
#I  number of polynomials after reduction is 42
#I  End of phase I
#I  End of phase II
#I  End of phase III
#I  End of phase IV
#I  The computation took 20 msecs.
]]></Listing>

The record GBR has two members: the two-sided relations <C>GBR.ts</C> and the
prefix relations <C>GBR.p</C>. We print these using the
function <Ref Func="PrintNPList" Style="Text"/>:

<Listing><![CDATA[
gap> PrintNPList(GBR.ts);
s1^2 - 1
s2^2 - 1
s3s1 - s1s3
s3^2 - 1
es1 - s1e
es2 - s2e
e^2 - e
s2s1s2 - s1s2s1
s3s2s3 - s2s3s2
s3s2s1s3 - s2s3s2s1
s3es3e - es3e
es3es3 - es3e
s3es3s2e - es3s2e
s2s3s2es3e - s3s2es3e
s3es3s2s1e - es3s2s1e
es3s2es3s2 - es3s2es3
s2s3s2s1es3e - s3s2s1es3e
s2s3s2es3s2e - s3s2es3s2e
s2es3s2es3e - es3s2es3e
s1s2s1s3s2es3e - s2s1s3s2es3e
s2s3s2s1es3s2e - s3s2s1es3s2e
s2s3s2es3s2s1e - s3s2es3s2s1e
s2es3s2s1es3e - es3s2s1es3e
es3s2s1es3s2s1 - es3s2s1es3s2
s1s2s1s3s2s1es3e - s2s1s3s2s1es3e
s1s2s1s3s2es3s2e - s2s1s3s2es3s2e
s1s2s1es3s2es3e - s2s1es3s2es3e
s2s3s2s1es3s2s1e - s3s2s1es3s2s1e
s2es3s2s1es3s2e - es3s2s1es3s2e
s1s2s1s3s2s1es3s2e - s2s1s3s2s1es3s2e
s1s2s1s3s2es3s2s1e - s2s1s3s2es3s2s1e
s1s2s1es3s2s1es3e - s2s1es3s2s1es3e
s1s3s2s1es3s2es3e - s3s2s1es3s2es3e
s1s2s1s3s2s1es3s2s1e - s2s1s3s2s1es3s2s1e
s1s2s1es3s2s1es3s2e - s2s1es3s2s1es3s2e
s1s3s2s1es3s2s1es3e - s3s2s1es3s2s1es3e
s1es3s2s1es3s2es3e - es3s2s1es3s2es3e
s1s3s2s1es3s2s1es3s2e - s3s2s1es3s2s1es3s2e
gap> PrintNPList(GBR.p);
[ s1 - 1 ]
[ s2 - 1 ]
[ e ]
[ s3e - s3 ]
[ s3s2e - s3s2 ]
[ s3s2s1e - s3s2s1 ]
]]></Listing>

It is now possible to calculate the standard basis of the quotient module
with the function <Ref Func="BaseQM" Style="Text"/>. This function has as
arguments the Gröbner basis record <C>GBR</C>, the number of generators of
the algebra (here this is 4), the number of generators of
the module (here this is 1),  and
a variable <C>maxno</C> for returning partial bases (0 means full
basis).

<Listing><![CDATA[
gap> B:=BaseQM(GBR,4,1,0);;
gap> PrintNPList(B);
[ 1 ]
[ s3 ]
[ s3s2 ]
[ s3s2s1 ]
]]></Listing>

Next we write down the matrices for the right action of the generators
on the module. First by means of the list command
<Ref Func="MatricesQA" Style="Text"/>, next one by one by means of
<Ref Func="MatrixQA" Style="Text"/> within a loop.

<Listing><![CDATA[
gap> MatricesQA(4,B,GBR);
[ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ] ],
  [ [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ] ],
  [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ],
  [ [ 0, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ] ]
gap> for i in [1..4] do
>   Display(MatrixQA(i,B,GBR)); Print("\n");
> od;
[ [  1,  0,  0,  0 ],
  [  0,  1,  0,  0 ],
  [  0,  0,  0,  1 ],
  [  0,  0,  1,  0 ] ]

[ [  1,  0,  0,  0 ],
  [  0,  0,  1,  0 ],
  [  0,  1,  0,  0 ],
  [  0,  0,  0,  1 ] ]

[ [  0,  1,  0,  0 ],
  [  1,  0,  0,  0 ],
  [  0,  0,  1,  0 ],
  [  0,  0,  0,  1 ] ]

[ [  0,  0,  0,  0 ],
  [  0,  1,  0,  0 ],
  [  0,  0,  1,  0 ],
  [  0,  0,  0,  1 ] ]

]]></Listing>

</Section>
<#/GAPDoc>

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