gap> START_TEST("GBNP test06");
gap> ######################### BEGIN COPYRIGHT MESSAGE #########################
gap> # GBNP - computing Gröbner bases of noncommutative polynomials
gap> # Copyright 2001-2010 by Arjeh M. Cohen, Dié A.H. Gijsbers, Jan Willem
gap> # Knopper, Chris Krook. Address: Discrete Algebra and Geometry (DAM) group
gap> # at the Department of Mathematics and Computer Science of Eindhoven
gap> # University of Technology.
gap> #
gap> # For acknowledgements see the manual. The manual can be found in several
gap> # formats in the doc subdirectory of the GBNP distribution. The
gap> # acknowledgements formatted as text can be found in the file chap0.txt.
gap> #
gap> # GBNP is free software; you can redistribute it and/or modify it under
gap> # the terms of the Lesser GNU General Public License as published by the
gap> # Free Software Foundation (FSF); either version 2.1 of the License, or
gap> # (at your option) any later version. For details, see the file 'LGPL' in
gap> # the doc subdirectory of the GBNP distribution or see the FSF's own site:
gap> #
https://www.gnu.org/licenses/lgpl.html
gap> ########################## END COPYRIGHT MESSAGE ##########################
gap>
gap> ### filename = "example06.g"
gap> ### authors Cohen & Gijsbers
gap>
gap> ### THIS IS A GAP PACKAGE FOR COMPUTING NON-COMMUTATIVE GROBNER BASES
gap>
gap> ### Last change: August 22 2001.
gap> ### amc
gap>
gap> # <#GAPDoc Label="Example06">
gap> # <Section Label="Example06"><Heading>From the Tapas book</Heading>
gap> # This example is a standard commutative Gröbner basis computation from the book
gap> # Some Tapas of Computer Algebra
gap> # <Cite Key="CohenCuypersSterk1999"/>, page 339.
gap> # There are six variables, named <M>a</M>, ... , <M>f</M> by default.
gap> # We work over the rationals and study the ideal generated by the twelve polynomials
gap> # occurring on the middle of page 339 of the Tapas book
gap> # in a project by De Boer and Pellikaan on the ternary cyclic code of length 11.
gap> # Below these are named <C>p1</C>, ..., <C>p12</C>.
gap> # The result should be the union of <M>\{a,b\}</M> and
gap> # the set of 6 homogeneous binomials
gap> # (that is, polynomials with two terms) of degree 2 forcing
gap> # commuting between <M>c</M>, <M>d</M>, <M>e</M>, and <M>f</M>.
gap> # <P/>
gap>
gap> # <!--
gap> # a = 1
gap> # b = 2
gap> # sigma_i = i+2 (i=1,2,3,4) = c,d,e,f -->
gap>
gap> # <P/>
gap> # First load the package and set the standard infolevel <Ref
gap> # InfoClass="InfoGBNP" Style="Text"/> to 2 and the time infolevel <Ref
gap> # Func="InfoGBNPTime" Style="Text"/> to 1 (for more information about the info
gap> # level, see Chapter <Ref Chap="Info"/>).
gap>
gap> # <L>
gap> LoadPackage("gbnp", false);
true
gap> SetInfoLevel(InfoGBNP,2);
gap> SetInfoLevel(InfoGBNPTime,0);
gap> # </L>
gap>
gap> # Now define some functions which will help in the construction of relations.
gap> # The function <C>powermon(g, exp)</C> will return the monomial <M>g^{exp}</M>.
gap> # The function <C>comm(a, b)</C> will return a relation forcing commutativity
gap> # between its two arguments <C>a</C> and <C>b</C>.
gap>
gap> # <L>
gap> powermon := function(base, exp)
> local ans,i;
> ans := [];
> for i in [1..exp] do ans := Concatenation(ans,[base]); od;
> return ans;
> end;;
gap>
gap> comm := function(a,b)
> return [[[a,b],[b,a]],[1,-1]];
> end;;
gap> # </L>
gap>
gap> # Now the relations are entered.
gap>
gap> # <L>
gap> p1 := [[[5,1]],[1]];;
gap> p2 := [[powermon(1,3),[6,1]],[1,1]];;
gap> p3 := [[powermon(1,9),Concatenation([3],powermon(1,3))],[1,1]];;
gap> p4 := [[powermon(1,81),Concatenation([3],powermon(1,9)),
> Concatenation([4],powermon(1,3))],[1,1,1]];;
gap> p5 := [[Concatenation([3],powermon(1,81)),Concatenation([4],powermon(1,9)),
> Concatenation([5],powermon(1,3))],[1,1,1]];;
gap> p6 := [[powermon(1,27),Concatenation([4],powermon(1,81)),Concatenation([5],
> powermon(1,9)),Concatenation([6],powermon(1,3))],[1,1,1,1]];;
gap> p7 := [[powermon(2,1),Concatenation([3],powermon(1,27)),Concatenation([5],
> powermon(1,81)),Concatenation([6],powermon(1,9))],[1,1,1,1]];;
gap> p8 := [[Concatenation([3],powermon(2,1)),Concatenation([4],powermon(1,27)),
> Concatenation([6],powermon(1,81))],[1,1,1]];;
gap> p9 := [[Concatenation([],powermon(1,1)),Concatenation([4],powermon(2,1)),
> Concatenation([5],powermon(1,27))],[1,1,1]];;
gap> p10 := [[Concatenation([3],powermon(1,1)),Concatenation([5],powermon(2,1)),
> Concatenation([6],powermon(1,27))],[1,1,1]];;
gap> p11 := [[Concatenation([4],powermon(1,1)),Concatenation([6],powermon(2,1))],
> [1,1]];;
gap> p12 := [[Concatenation([],powermon(2,3)),Concatenation([],powermon(2,1))],
> [1,-1]];;
gap> KI := [p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12];;
gap> for i in [1..5] do
> for j in [i+1..6] do
> Add(KI,comm(i,j));
> od;
> od;
gap> # </L>
gap>
gap> # add the next command in case other tests have changed the alphabet:
gap> GBNP.ConfigPrint("a","b","c","d","e","f");
gap> # The relations can be shown with <Ref Func="PrintNPList" Style="Text"/>:
gap>
gap> # <L>
gap> PrintNPList(KI);
ea
a^3 + fa
a^9 + ca^3
a^81 + ca^9 + da^3
ca^81 + da^9 + ea^3
a^27 + da^81 + ea^9 + fa^3
b + ca^27 + ea^81 + fa^9
cb + da^27 + fa^81
a + db + ea^27
ca + eb + fa^27
da + fb
b^3 - b
ab - ba
ac - ca
ad - da
ae - ea
af - fa
bc - cb
bd - db
be - eb
bf - fb
cd - dc
ce - ec
cf - fc
de - ed
df - fd
ef - fe
gap> Length(KI);
27
gap> # </L>
gap>
gap> # It is sometimes easier to enter the relations as elements of a free algebra
gap> # and then use the function <Ref Func="GP2NP" Style="Text"/> or the function
gap> # <Ref Func="GP2NPList" Style="Text"/> to convert them.
gap> # This will be demonstrated below. More about converting can be read
gap> # in Section <Ref Sect="TransitionFunctions"/>.
gap>
gap> # <L>
gap> F:=Rationals;;
gap> A:=FreeAssociativeAlgebraWithOne(F,"a","b","c","d","e","f");;
gap> a:=A.a;; b:=A.b;; c:=A.c;; d:=A.d;; e:=A.e;; f:=A.f;;
gap> KI_gp:=[e*a, #p1
> a^3 + f*a, #p2
> a^9 + c*a^3, #p3
> a^81 + c*a^9 + d*a^3, #p4
> c*a^81 + d*a^9 + e*a^3, #p5
> a^27 + d*a^81 + e*a^9 + f*a^3, #p6
> b + c*a^27 + e*a^81 + f*a^9, #p7
> c*b + d*a^27 + f*a^81, #p8
> a + d*b + e*a^27, #p9
> c*a + e*b + f*a^27, #p10
> d*a + f*b, #p11
> b^3 - b];; #p12
gap> # </L>
gap>
gap> # These relations can be converted to NP form (see <Ref Sect="NP"/>) with <Ref
gap> # Func="GP2NPList" Style="Text"/>. For use in a Gröbner basis computation we have to
gap> # order the NP polynomials in <C>KI</C>.
gap> # This can be done with <Ref Func="CleanNP" Style="Text"/>.
gap>
gap> # <L>
gap> KI_np:=GP2NPList(KI_gp);;
gap> Apply(KI,x->CleanNP(x));;
gap> KI_np=KI{[1..12]};
true
gap> # </L>
gap>
gap> # The Gröbner basis can now be calculated with <Ref Func="SGrobner"
gap> # Style="Text"/> and printed with <Ref Func="PrintNPList" Style="Text"/>.
gap>
gap> # <L>
gap> GB := SGrobner(KI);;
#I number of entered polynomials is 27
#I number of polynomials after reduction is 8
#I End of phase I
#I End of phase II
#I List of todo lengths is [ 0 ]
#I End of phase III
#I G: Cleaning finished, 0 polynomials reduced
#I End of phase IV
gap> PrintNPList(GB);
a
b
dc - cd
ec - ce
ed - de
fc - cf
fd - df
fe - ef
gap> # </L>
gap>
gap>
gap> # </Section>
gap> # <#/GAPDoc>
gap>
gap> STOP_TEST("test06.g",10000);
