Groups Defined by Rewrite Systems

• Introduction
  • Terminology
  • The Category of Rewrite Groups
  • The Construction of a Rewrite Group
• Constructing Confluent Presentations
  • The Knuth-Bendix Procedure
    • RWSGroup(F: parameters): MonFP → MonRWS
    • Example: Rewrite Group for the Von Dyck (2,3,5) Group
  • Defining Orderings
    • RWSGroup(F: parameters): MonFP → MonRWS
    • Example: Recursive Ordering for a Non-Hopfian Group
    • Example: Rewrite System for a Free Nilpotent Group
  • Setting Limits
    • RWSMonoid(F: parameters): MonFP → MonRWS
    • SetVerbose("KBMAG", v): MonStgElt, RngIntElt
  • Accessing Group Information
    • G . i: GrpRWS, RngIntElt → GrpRWSElt
    • Generators(G): GrpRWS → [GrpRWSElt]
    • NumberOfGenerators(G): GrpRWS → RngIntElt
    • Ngens(G): GrpRWS → RngIntElt
    • Relations(G): GrpRWS → [GrpFPRel]
    • NumberOfRelations(G): GrpRWS → RngIntElt
    • Nrels(G): GrpRWS → RngIntElt
    • Ordering(G): GrpRWS → String
    • Example: Accessing Rewrite Group Information
• Properties of a Rewrite Group
  • IsConfluent(G): GrpRWS → BoolElt
  • IsFinite(G): GrpRWS → BoolElt, RngIntElt
  • Order(G): GrpRWS → RngIntElt
  • # G: GrpRWS → RngIntElt
  • Example: Confluence of the Weyl Group E8
  • Example: Order of an Infinite Rewrite Group
• Arithmetic with Words
  • Construction of a Word
    • Identity(G): GrpRWS → GrpRWSElt
    • Id(G): GrpRWS → GrpRWSElt
    • G ! 1: GrpRWS, RngIntElt → GrpRWSElt
    • G ! [ i₁, ..., iₛ ]: GrpRWS, [ RngIntElt ] → GrpRWSElt
    • Parent(w): GrpRWSElt → GrpRWS
    • Example: Constructing Words in a Rewrite Group
  • Element Operations
    • u * v: GrpRWSElt, GrpRWSElt → GrpRWSElt
    • u / v: GrpRWSElt, GrpRWSElt → GrpRWSElt
    • u ^ n: GrpRWSElt, RngIntElt → GrpRWSElt
    • u ^ v: GrpRWSElt, GrpRWSElt → GrpRWSElt
    • Inverse(w): GrpRWSElt → GrpRWSElt
    • (u, v): GrpRWSElt, GrpRWSElt → GrpRWSElt
    • (u₁, ..., uᵣ): GrpRWSElt, ..., GrpRWSElt → GrpRWSElt
    • u eq v: GrpRWSElt, GrpRWSElt → BoolElt
    • u ne v: GrpRWSElt, GrpRWSElt → BoolElt
    • IsId(w): GrpRWSElt → BoolElt
    • IsIdentity(w): GrpRWSElt → BoolElt
    • # u: GrpRWSElt → RngIntElt
    • ElementToSequence(u): GrpRWSElt → [ RngIntElt ]
    • Eltseq(u): GrpRWSElt → [ RngIntElt ]
    • Example: Word Arithmetic in a Fibonacci Group
• Operations on the Set of Group Elements
  • Random(G, n): GrpRWS, RngIntElt → GrpRWSElt
  • Random(G): GrpRWS → GrpRWSElt
  • Representative(G): GrpRWS → GrpRWSElt
  • Rep(G): GrpRWS → GrpRWSElt
  • Set(G, a, b): GrpRWS, RngIntElt, RngIntElt → SetEnum
  • Set(G): GrpRWS → SetEnum
  • Seq(G, a, b): GrpRWS, RngIntElt, RngIntElt → SeqEnum
  • Seq(G): GrpRWS → SeqEnum
  • Example: Element Sets of a Rewrite Group
• Homomorphisms
  • General Remarks
  • Construction of Homomorphisms
    • hom< R -> G | S >: Struct, Struct → Map
• Conversion to a Finitely Presented Group
  • Example: Converting a Rewrite Group to an FP-Group
• Bibliography