  
  
                       [1XDocumentation on the [5XGBNP[105X package[101X
  
  
                                 Version 1.0.3
  
  
                                  8 March 2016
  
  
                                 Arjeh M. Cohen
  
                               Jan Willem Knopper
  
  
  
  Arjeh M. Cohen
      Email:    [7Xmailto:A.M.Cohen@tue.nl[107X
  Jan Willem Knopper
      Email:    [7Xmailto:J.W.Knopper@tue.nl[107X
  
  
  Address: [33X[0;9YTU/e,[133X
           [33X[0;9YPOB 513, 5600 MB Eindhoven, the Netherlands[133X
  
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YWe  provide  algorithms,  written  in  the  [5XGAP[105X  4 programming language, for
  computing Gröbner bases of non-commutative polynomials, and some variations,
  such  as  a  weighted  and  truncated  version  and  a  tracing facility. In
  addition,   there   are   algorithms   for   analyzing  the  quotient  of  a
  non-commutative  polynomial algebra by a 2-sided ideal generated by a set of
  polynomials  whose  Gröbner  basis  has  been  determined  and for computing
  quotient modules of free modules over quotient algebras.[133X
  
  [33X[0;0YThe notion of algorithm is interpreted loosely: in general one cannot expect
  a  non-commutative  Gröbner  basis algorithm to terminate, as it would imply
  solvability of the word problem for finitely presented (semi)groups.[133X
  
  [33X[0;0YThis  documentation  gives  a short description of the mathematical content,
  explains  the functions of the package, and provides more than twenty worked
  out examples.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [30X    [33X[0;6YThe package is based on an earlier version by Rosane Ushirobira.[133X
  
  [30X    [33X[0;6YThe  bulk  of  the  package  is written by Arjeh M. Cohen and Dié A.H.
        Gijsbers.[133X
  
  [30X    [33X[0;6YThe  theory  is  mainly  taken from literature by Teo Mora [Mor94] and
        Edward L. Green [Gre99].[133X
  
  [30X    [33X[0;6YFrom   Version  0.8.3  on  the  package  has  three  additional  files
        ([11Xfincheck.g[111X,  [11Xtree.g[111X  [11Xgraphs.g[111X)  with routines for finding the Hilbert
        function  and testing finite dimensionality when given a Gröbner basis
        by Chris Krook [Kro03], based on work by Victor Ufnarovski [Ufn89].[133X
  
  [30X    [33X[0;6YFrom  Version  0.9  on the package is enriched with support for fields
        implemented  in  GAP and additional prefix rules for quotient modules,
        as  well as some speed improvements by Jan Willem Knopper. Knopper has
        also formatted the documentation in GAPDoc [LN06].[133X
  
  [30X    [33X[0;6YFrom   Version   1.0   on  the  package  is  extended  with  NMO  (for
        Noncommutative  Monomial  Orderings) by Randall Cone. This enables the
        GBNP  user  to choose a wider selection of monomial orderings than the
        standard one built into GBNP itself. Documentation on NMO can be found
        in the NMO manual [Con10].[133X
  
  [33X[0;0Y [133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (GBNP)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YInstallation[133X
    1.2 [33X[0;0YUsing the package[133X
    1.3 [33X[0;0YFurther documentation[133X
  2 [33X[0;0YDescription[133X
    2.1 [33X[0;0YNon-commutative Polynomials (NPs)[133X
    2.2 [33X[0;0YNon-commutative Polynomials for Modules (NPMs)[133X
    2.3 [33X[0;0YCore functions[133X
    2.4 [33X[0;0YAbout the implementation[133X
    2.5 [33X[0;0YTracing variant[133X
    2.6 [33X[0;0YTruncation variant[133X
    2.7 [33X[0;0YModule variant[133X
    2.8 [33X[0;0YGröbner basis records[133X
    2.9 [33X[0;0YQuotient algebras[133X
  3 [33X[0;0YFunctions[133X
    3.1 [33X[0;0YConverting polynomials into different formats[133X
      3.1-1 GP2NP
      3.1-2 GP2NPList
      3.1-3 NP2GP
      3.1-4 NP2GPList
    3.2 [33X[0;0YPrinting polynomials in NP format[133X
      3.2-1 PrintNP
      3.2-2 GBNP.ConfigPrint
      3.2-3 PrintNPList
    3.3 [33X[0;0YCalculating with polynomials in NP format[133X
      3.3-1 NumAlgGensNP
      3.3-2 NumAlgGensNPList
      3.3-3 NumModGensNP
      3.3-4 NumModGensNPList
      3.3-5 AddNP
      3.3-6 BimulNP
      3.3-7 CleanNP
      3.3-8 GtNP
      3.3-9 LtNP
      3.3-10 LMonsNP
      3.3-11 MkMonicNP
      3.3-12 MulNP
    3.4 [33X[0;0YGröbner functions, standard variant[133X
      3.4-1 Grobner
      3.4-2 SGrobner
      3.4-3 IsGrobnerBasis
      3.4-4 IsStrongGrobnerBasis
      3.4-5 IsGrobnerPair
      3.4-6 MakeGrobnerPair
    3.5 [33X[0;0YFinite-dimensional quotient algebras[133X
      3.5-1 BaseQA
      3.5-2 DimQA
      3.5-3 MatrixQA
      3.5-4 MatricesQA
      3.5-5 MulQA
      3.5-6 StrongNormalFormNP
    3.6 [33X[0;0YFiniteness and Hilbert series[133X
      3.6-1 DetermineGrowthQA
      3.6-2 FinCheckQA
      3.6-3 HilbertSeriesQA
      3.6-4 PreprocessAnalysisQA
    3.7 [33X[0;0YFunctions of the trace variant[133X
      3.7-1 EvalTrace
      3.7-2 PrintTraceList
      3.7-3 PrintTracePol
      3.7-4 PrintNPListTrace
      3.7-5 SGrobnerTrace
      3.7-6 StrongNormalFormTraceDiff
    3.8 [33X[0;0YFunctions of the truncated variant[133X
      3.8-1 [33X[0;0YExamples[133X
      3.8-2 SGrobnerTrunc
      3.8-3 CheckHomogeneousNPs
      3.8-4 BaseQATrunc
      3.8-5 DimsQATrunc
      3.8-6 FreqsQATrunc
    3.9 [33X[0;0YFunctions of the module variant[133X
      3.9-1 SGrobnerModule
      3.9-2 BaseQM
      3.9-3 DimQM
      3.9-4 MulQM
      3.9-5 StrongNormalFormNPM
  4 [33X[0;0YInfo Level[133X
    4.1 [33X[0;0YIntroduction[133X
    4.2 [33X[0;0YInfoGBNP[133X
      4.2-1 InfoGBNP
      4.2-2 [33X[0;0YWhat will be printed at level 0[133X
      4.2-3 [33X[0;0YWhat will be printed at level 1[133X
      4.2-4 [33X[0;0YWhat will be printed at level 2[133X
    4.3 [33X[0;0YInfoGBNPTime[133X
      4.3-1 InfoGBNPTime
      4.3-2 [33X[0;0YWhat will be printed at level 0[133X
      4.3-3 [33X[0;0YWhat will be printed at level 1[133X
      4.3-4 [33X[0;0YWhat will be printed at level 2[133X
  A [33X[0;0YExamples[133X
    A.1 [33X[0;0YIntroduction[133X
    A.2 [33X[0;0YA simple commutative Gröbner basis computation[133X
    A.3 [33X[0;0YA truncated Gröbner basis for Leonard pairs[133X
    A.4 [33X[0;0YThe truncated variant on two weighted homogeneous polynomials[133X
    A.5 [33X[0;0YThe order of the Weyl group of type E[22X_6[122X[133X
    A.6 [33X[0;0YThe gcd of some univariate polynomials[133X
    A.7 [33X[0;0YFrom the Tapas book[133X
    A.8 [33X[0;0YThe Birman-Murakami-Wenzl algebra of type A[22X_3[122X[133X
    A.9 [33X[0;0YThe Birman-Murakami-Wenzl algebra of type A[22X_2[122X[133X
    A.10 [33X[0;0YA commutative example by Mora[133X
    A.11 [33X[0;0YTracing an example by Mora[133X
    A.12 [33X[0;0YFiniteness of the Weyl group of type E[22X_6[122X[133X
    A.13 [33X[0;0YPreprocessing for Weyl group computations[133X
    A.14 [33X[0;0YA quotient algebra with exponential growth[133X
    A.15 [33X[0;0YA commutative quotient algebra of polynomial growth[133X
    A.16 [33X[0;0YAn algebra over a finite field[133X
    A.17 [33X[0;0YThe dihedral group of order 8[133X
    A.18 [33X[0;0YThe dihedral group of order 8 on another module[133X
    A.19 [33X[0;0YThe dihedral group on a non-cyclic module[133X
    A.20 [33X[0;0YThe icosahedral group[133X
    A.21 [33X[0;0YThe symmetric inverse monoid for a set of size four[133X
    A.22 [33X[0;0YA module of the Hecke algebra of type A[22X_3[122X over GF(3)[133X
    A.23 [33X[0;0YGeneralized Temperley-Lieb algebras[133X
    A.24 [33X[0;0YThe universal enveloping algebra of a Lie algebra[133X
    A.25 [33X[0;0YSerre's exercise[133X
    A.26 [33X[0;0YBaur and Draisma's transformations[133X
    A.27 [33X[0;0YThe cola gene puzzle[133X
  
  
  [32X
