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: http://www.gnu.org/licenses/lgpl.html gap> ########################## END COPYRIGHT MESSAGE ##########################

gap> LoadPackage("gbnp"); ───────────────────────────────────────────────────────────────────────────── Loading GBNP 1.0.3 (Non-commutative Gröbner bases) by A.M. Cohen (http://www.win.tue.nl/~amc) and J.W. Knopper (J.W.Knopper@tue.nl). Homepage: http://mathdox.org/products/gbnp/ ───────────────────────────────────────────────────────────────────────────── true

gap> NPzero:=[[],[]];; gap> NPone:=[[[]],[1]];; gap> NPone_2:=[[[]],[One(GF(2))]];; gap> NPMone1:=[[[-1,]],[1]];; gap> NPMone2:=[[[-2,]],[1]];; gap> NPMone1_2:=[[[-1,]],[One(GF(2))]];; gap> NPMone2_2:=[[[-2,]],[One(GF(2))]];;

gap> GBRempty:=SGrobnerModule([],[]);; gap> GBRone:=SGrobnerModule([],[NPone]);; gap> GBRone_12:=SGrobnerModule([NPMone1,NPMone2],[]);; gap> GBRone_1:=SGrobnerModule([NPMone1],[]);; BaseQM gap> BaseQM(GBRempty,1,1,2); "infinite" gap> BaseQM(GBRone,0,0,0)=[]; true gap> BaseQM(GBRone_12,0,0,0)=[]; true gap> BaseQM(GBRone_12,0,2,0)=[]; true

DimQM gap> DimQM(GBRone,0,0)=0; true gap> DimQM(GBRone_12,0,0)=0; true gap> DimQM(GBRone_12,0,2)=0; true

MulQM gap> MulQM(NPone,NPone,GBRempty)=NPone; true gap> MulQM(NPone,NPzero,GBRempty)=NPzero; true gap> MulQM(NPzero,NPone,GBRempty)=NPzero; true gap> MulQM(NPzero,NPzero,GBRempty)=NPzero; true gap> MulQM(NPone,NPone,GBRone)=NPzero; true gap> MulQM(NPMone1,NPone,GBRone_1)=NPzero; true gap> MulQM(NPMone2,NPone,GBRone_1)=NPMone2; true

SGrobnerModule check function gap> check:=function(r, p, ts) > return (r.p = p) and (r.ts = ts); > end;;

first some standard checks gap> check(SGrobnerModule([],[]), [], []); true gap> check(SGrobnerModule([],[NPzero]), [], []); true gap> check(SGrobnerModule([],[NPone]), [], [NPone]); true gap> check(SGrobnerModule([],[NPone_2]), [], [NPone_2]); true now some prefix checks gap> check(SGrobnerModule([NPMone1],[]),[NPMone1],[]); true gap> check(SGrobnerModule([NPMone1],[NPone]),[],[NPone]); true gap> check(SGrobnerModule([NPMone1_2],[NPone_2]),[],[NPone_2]); true gap> check(SGrobnerModule([NPMone1_2,NPMone2_2],[NPone_2]),[],[NPone_2]); true gap> check(SGrobnerModule([NPMone1_2,NPMone2_2],[]),[NPMone2_2,NPMone1_2],[]); true

StrongNormalFormNPM gap> StrongNormalFormNPM(NPzero,GBRempty)=NPzero; true gap> StrongNormalFormNPM(NPone,GBRempty)=NPone; true gap> StrongNormalFormNPM(NPMone1,GBRempty)=NPMone1; true gap> StrongNormalFormNPM(NPMone1,GBRone)=NPzero; true gap> StrongNormalFormNPM(NPone,GBRone)=NPzero; true gap> StrongNormalFormNPM(NPone,GBRone_12)=NPone; true gap> StrongNormalFormNPM(NPMone1,GBRone_12)=NPzero; true