  
  
                       [1XDocumentation on the [5XGBNP[0m[1X package[0m
  
  
                                 Version 1.0.1
  
  
                                  12 May 2010
  
  
                                 Arjeh M. Cohen
  
                               Jan Willem Knopper
  
  
  
  Arjeh M. Cohen
      Email:    [7Xmailto:A.M.Cohen@tue.nl[0m
  Jan Willem Knopper
      Email:    [7Xmailto:J.W.Knopper@tue.nl[0m
  
  
  Address: TU/e,
           POB 513, 5600 MB Eindhoven, the Netherlands
  
  
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  [1XAbstract[0m
  We  provide  algorithms,  written  in  the  [5XGAP[0m  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.
  
  The 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.
  
  This  documentation  gives  a short description of the mathematical content,
  explains  the functions of the package, and provides more than twenty worked
  out examples.
  
  
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  [1XAcknowledgements[0m
  --    The package is based on an earlier version by Rosane Ushirobira.
  
  --    The  bulk  of  the  package  is written by Arjeh M. Cohen and Dié A.H.
        Gijsbers.
  
  --    The  theory  is  mainly  taken from literature by Teo Mora [Mor94] and
        Edward L. Green [Gre99].
  
  --    From   Version  0.8.3  on  the  package  has  three  additional  files
        ([11Xfincheck.g[0m,  [11Xtree.g[0m  [11Xgraphs.g[0m)  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].
  
  --    From  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].
  
  --    From   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].
  
   
  
  
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  [1XContents (GBNP)[0X
  
  1 Introduction
    1.1 Installation
    1.2 Using the package
    1.3 Further documentation
  2 Description
    2.1 Non-commutative Polynomials (NPs)
    2.2 Non-commutative Polynomials for Modules (NPMs)
    2.3 Core functions
    2.4 About the implementation
    2.5 Tracing variant
    2.6 Truncation variant
    2.7 Module variant
    2.8 Gröbner basis records
    2.9 Quotient algebras
  3 Functions
    3.1 Converting polynomials into different formats
      3.1-1 GP2NP
      3.1-2 GP2NPList
      3.1-3 NP2GP
      3.1-4 NP2GPList
    3.2 Printing polynomials in NP format
      3.2-1 PrintNP
      3.2-2 GBNP.ConfigPrint
      3.2-3 PrintNPList
    3.3 Calculating with polynomials in NP format
      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 Gröbner functions, standard variant
      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 Finite-dimensional quotient algebras
      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 Finiteness and Hilbert series
      3.6-1 DetermineGrowthQA
      3.6-2 FinCheckQA
      3.6-3 HilbertSeriesQA
      3.6-4 PreprocessAnalysisQA
    3.7 Functions of the trace variant
      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 Functions of the truncated variant
      3.8-1 Examples
      3.8-2 SGrobnerTrunc
      3.8-3 CheckHomogeneousNPs
      3.8-4 BaseQATrunc
      3.8-5 DimsQATrunc
      3.8-6 FreqsQATrunc
    3.9 Functions of the module variant
      3.9-1 SGrobnerModule
      3.9-2 BaseQM
      3.9-3 DimQM
      3.9-4 MulQM
      3.9-5 StrongNormalFormNPM
  4 Info Level
    4.1 Introduction
    4.2 InfoGBNP
      4.2-1 InfoGBNP
      4.2-2 What will be printed at level 0
      4.2-3 What will be printed at level 1
      4.2-4 What will be printed at level 2
    4.3 InfoGBNPTime
      4.3-1 InfoGBNPTime
      4.3-2 What will be printed at level 0
      4.3-3 What will be printed at level 1
      4.3-4 What will be printed at level 2
  A. Examples
    A.1 Introduction
    A.2 A simple commutative Gröbner basis computation
    A.3 A truncated Gröbner basis for Leonard pairs
    A.4 The truncated variant on two weighted homogeneous polynomials
    A.5 The order of the Weyl group of type E_6
    A.6 The gcd of some univariate polynomials
    A.7 From the Tapas book
    A.8 The Birman-Murakami-Wenzl algebra of type A_3
    A.9 The Birman-Murakami-Wenzl algebra of type A_2
    A.10 A commutative example by Mora
    A.11 Tracing an example by Mora
    A.12 Finiteness of the Weyl group of type E_6
    A.13 Preprocessing for Weyl group computations
    A.14 A quotient algebra with exponential growth
    A.15 A commutative quotient algebra of polynomial growth
    A.16 An algebra over a finite field
    A.17 The dihedral group of order 8
    A.18 The dihedral group of order 8 on another module
    A.19 The dihedral group on a non-cyclic module
    A.20 The icosahedral group
    A.21 The symmetric inverse monoid for a set of size four
    A.22 A module of the Hecke algebra of type A_3 over GF(3)
    A.23 Generalized Temperley-Lieb algebras
    A.24 The universal enveloping algebra of a Lie algebra
    A.25 Serre's exercise
    A.26 Baur and Draisma's transformations
    A.27 The cola gene puzzle
  
  
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