  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  [33X[0;0YThis  package, named GBNP for Gröbner Bases for Non-commutative Polynomials,
  is  intended  for computing in (associative) non-commutative algebras with a
  finite  presentation.  Starting  from a free algebra [22XA[122X on a finite number of
  generating  variables,  the reader can specify a finite set [22XG[122X of polynomials
  in  these  variables,  in  order  to  study the quotient algebra of [22XA[122X by the
  (2-sided) ideal of [22XA[122X generated by [22XG[122X.[133X
  
  [33X[0;0YThis  documentation gives a short description of the mathematical content in
  Chapter  [14X2[114X, explains the functions of the package in Chapter [14X3[114X, and provides
  more  than twenty four worked out examples in Appendix [14XA[114X. It is available as
  an HTML document at [7Xhttp://mathdox.org/products/gbnp/[107X and as an pdf document
  at [7Xhttp://mathdox.org/products/gbnp/manual.pdf[107X.[133X
  
  
  [1X1.1 [33X[0;0YInstallation[133X[101X
  
  [33X[0;0YTo        install       GBNP,       first       download       it       from
  [7Xhttp://mathdox.org/products/gbnp/GBNP-1.0.3.tar.gz[107X,        then       unpack
  [11XGBNP-1.0.3.tar.gz[111X  in  the  [10Xpkg[110X subdirectory of your [5XGAP[105X installation (or in
  the  [10Xpkg[110X subdirectory of any other [5XGAP[105X root directory, for example one added
  with   the   [10X-l[110X   argument)   with   the   following   command:   [10Xtar  -xvzf
  GBNP-1.0.3.tar.gz[110X.[133X
  
  [33X[0;0YGBNP is then loaded with the GAP command[133X
  
  [4X[32X[104X
    [4Xgap> LoadPackage( "GBNP" ); [104X
  [4X[32X[104X
  
  [33X[0;0YThose   who   want   to   download   this   documentation  can  find  it  at
  [7Xhttp://mathdox.org/products/gbnp/GBNPdoc-1.0.3.tar.gz[107X  and  extract  it with
  [10Xtar -xvzf GBNPdoc-1.0.3.tar.gz[110X. It is also included in the package.[133X
  
  
  [1X1.2 [33X[0;0YUsing the package[133X[101X
  
  [33X[0;0YIf   you   wish   to   compute  a  Gröbner  basis,  create  a  list  of  NPs
  (non-commutative  polynomials  in  NP  format), as described in Section [14X2.1[114X.
  This  can  be  done  either  directly  or by use of the transition functions
  described  in  Section  [14X3.1[114X. To run the standard algorithm use the functions
  from  Section  [14X3.4[114X.  With  these  functions,  you can try and find a Gröbner
  basis.  The word try is included because the algorithm for computing Gröbner
  bases  is not guaranteed to terminate. Printing issues for polynomials in NP
  format  are  discussed in Section [14X3.2[114X. If the Gröbner basis is found and the
  dimension  of  the  quotient  algebra [22XQ[122X (see Section [14X2.9[114X) is finite, you can
  find  a  basis  of  monomials for [22XQ[122X with the functions in Section [14X3.5[114X. For a
  more  advanced  analysis  of  [22XQ[122X,  such  as  a  proof  of  finite or infinite
  dimensionality, or for determining its growth or its partial Hilbert series,
  use the functions from Section [14X3.6[114X .[133X
  
  [33X[0;0YThere  are  three  variants  of  the  Gröbner basis algorithm, the truncated
  version,  the  trace  version,  and  the  module  version. In the (weighted)
  homogeneous case (described in Section [14X2.6[114X), the truncated version, given by
  the functions described in Section [14X3.8[114X, computes the part of a Gröbner basis
  up  to  an  indicated  weight. The trace version (described in Section [14X2.5[114X),
  given  by  the functions described in Section [14X3.7[114X, computes an expression of
  the  polynomials  of  the  Gröbner  basis  found  in  terms  of the original
  generators.  The  module  version (described in Sections [14X2.2[114X, [14X2.7[114X, and [14X2.8[114X),
  given  by  the  functions described in Section [14X3.9[114X, computes a Gröbner basis
  for a submodule of a free [22XQ[122X-module of finite rank.[133X
  
  [33X[0;0YRead the example files in Chapter [14XA[114X for inspiration. The source of the files
  can  be  perused  for  auxiliary functions, which are often used in the main
  functions but not deemed necessary for a first time user.[133X
  
  
  [1X1.3 [33X[0;0YFurther documentation[133X[101X
  
  [33X[0;0YThe  reports [Coh07], [Kro03], and [Kno04] can be downloaded from the web at
  these addresses:[133X
  
  [33X[0;0YThe  report [21XNon-commutative polynomial computations[121X, by Arjeh M. Cohen (with
  support  of  Dié  Gijsbers,  Jan  Willem  Knopper,  and  Chris Krook) can be
  downloaded from [7Xhttp://mathdox.org/products/gbnp/gbnp.pdf[107X.[133X
  
  [33X[0;0YThe  report  [21XDimensionality  of  quotient  algebras[121X,  by  Chris Krook can be
  downloaded from [7Xhttp://mathdox.org/products/gbnp/dqa.pdf[107X.[133X
  
  [33X[0;0YThe  report  [21XGBNP  and  vector  enumeration[121X,  by  Jan  Willem Knopper can be
  downloaded from [7Xhttp://mathdox.org/products/gbnp/knopper.pdf[107X.[133X
  
