GROUPS DEFINED BY REWRITE SYSTEMS

DETAILS

 
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 GrpRWS_RWSGroup (H34E1)

      Defining Orderings
            RWSGroup(F: parameters) : MonFP -> MonRWS
            Example GrpRWS_RWSGroup-2 (H34E2)
            Example GrpRWS_RWSGroup-3 (H34E3)

      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
            Relations(G) : GrpRWS -> [GrpFPRel]
            NumberOfRelations(G) : GrpRWS -> RngIntElt
            Ordering(G) : GrpRWS -> String
            Example GrpRWS_BasicAccess (H34E4)

 
Properties of a Rewrite Group
      IsConfluent(G) : GrpRWS -> BoolElt
      IsFinite(G) : GrpRWS -> BoolElt, RngIntElt
      Order(G) : GrpRWS -> RngIntElt
      Example GrpRWS_IsConfluent (H34E5)
      Example GrpRWS_Order (H34E6)

 
Arithmetic with Words

      Construction of a Word
            Identity(G) : GrpRWS -> GrpRWSElt
            G ! [ i_1, ..., i_s ] : GrpRWS, [ RngIntElt ] -> GrpRWSElt
            Parent(w) : GrpRWSElt -> GrpRWS
            Example GrpRWS_Words (H34E7)

      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
            (u_1, ..., u_r) : GrpRWSElt, ..., GrpRWSElt -> GrpRWSElt
            u eq v : GrpRWSElt, GrpRWSElt -> BoolElt
            u ne v : GrpRWSElt, GrpRWSElt -> BoolElt
            IsId(w) : GrpRWSElt -> BoolElt
            # u : GrpRWSElt -> RngIntElt
            ElementToSequence(u) : GrpRWSElt -> [ RngIntElt ]
            Example GrpRWS_Arithmetic (H34E8)

 
Operations on the Set of Group Elements
      Random(G, n) : GrpRWS, RngIntElt -> GrpRWSElt
      Random(G) : GrpRWS -> GrpRWSElt
      Representative(G) : GrpRWS -> GrpRWSElt
      Set(G, a, b) : GrpRWS, RngIntElt, RngIntElt -> SetEnum
      Set(G) : GrpRWS -> SetEnum
      Seq(G, a, b) : GrpRWS, RngIntElt, RngIntElt -> SeqEnum
      Example GrpRWS_Set (H34E9)

 
Homomorphisms

      General Remarks

      Construction of Homomorphisms
            hom< R -> G | S > : Struct , Struct -> Map

 
Conversion to a Finitely Presented Group

 
Bibliography