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FINITELY PRESENTED SEMIGROUPS

 
Acknowledgements
 
Introduction
 
The Construction of Free Semigroups and their Elements
      Structure Constructors
      Element Constructors
 
Elementary Operators for Words
      Multiplication and Exponentiation
      The Length of a Word
      Equality and Comparison
 
Specification of a Presentation
      Relations
      Presentations
      Accessing the Defining Generators and Relations
 
Subsemigroups, Ideals and Quotients
      Subsemigroups and Ideals
      Quotients
 
Extensions
 
Elementary Tietze Transformations
 
String Operations on Words







DETAILS

 
Introduction

 
The Construction of Free Semigroups and their Elements

      Structure Constructors
            FreeSemigroup(n) : RngIntElt -> SgpFP
            FreeMonoid(n) : RngIntElt -> MonFP
            Example SgpFP_FreeSemigroup (H77E1)

      Element Constructors
            S ! [i1, ... is] : SgpFP, [RngIntElt] -> SgpFPElt
            Id(M) : MonFP -> MonFPElt

 
Elementary Operators for Words

      Multiplication and Exponentiation
            u * v : SgpFPElt, SgpFPElt -> SgpFPElt
            u ^ n : SgpFPElt, RngIntElt -> SgpFPElt
            G ! Q : SgpFP, [ SgpFPElt ] -> SgpFPElt

      The Length of a Word
            # u : SgpFPElt -> RngIntElt

      Equality and Comparison
            u eq v : SgpFPElt, SgpFPElt -> BoolElt
            u ne v : SgpFPElt, SgpFPElt -> BoolElt
            u lt v : SgpFPElt, SgpFPElt -> BoolElt
            u le v : SgpFPElt, SgpFPElt -> BoolElt
            u ge v : SgpFPElt, SgpFPElt -> BoolElt
            u gt v : SgpFPElt, SgpFPElt -> BoolElt
            IsOne(u) : MonFPElt -> BoolElt

 
Specification of a Presentation

      Relations
            w1 = w2 : SgpFPElt, SgpFPElt -> Rel
            LHS(r) : Rel -> SgpFPElt
            RHS(r) : Rel -> SgpFPElt

      Presentations
            Semigroup< generators | relations > : SgpFPElt, ..., SgpFPElt, Rel, ...Rel -> SgpFP
            Monoid< generators | relations > : MonFPElt, ..., MonFPElt, Rel, ..., Rel -> MonFP
            Example SgpFP_Monoid (H77E2)

      Accessing the Defining Generators and Relations
            S . i : SgpFP, RngIntElt -> SgpFPElt
            Generators(S) : SgpFP -> { SgpFPElt }
            NumberOfGenerators(S) : SgpFP -> RngIntElt
            Parent(u) : SgpFPElt -> SgpFP
            Relations(S) : SgpFP -> [ Rel ]

 
Subsemigroups, Ideals and Quotients

      Subsemigroups and Ideals
            sub<S | L1, ..., Lr> : SgpFP, SgpFPElt, ..., SgpFPElt -> SgpFP
            ideal<S | L1, ..., Lr> : SgpFP, SgpFPElt, ..., SgpFPElt -> SgpFPIdl
            lideal<G | L1, ..., Lr> : SgpFP, SgpFPElt, ..., SgpFPElt -> SgpFPIdl
            rideal<G | L1, ..., Lr> : SgpFP, SgpFPElt, ..., SgpFPElt -> SgpFPIdl

      Quotients
            quo< F | relations > : SgpFP, Rel, ..., Rel -> SgpFP

 
Extensions
      DirectProduct(R, S) : SgpFP, SgpFP -> SgpFP
      FreeProduct(R, S) : SgpFP, SgpFP -> SgpFP

 
Elementary Tietze Transformations
      AddRelation(S, r) : SgpFP, Rel -> SgpFP
      DeleteRelation(S, r) : SgpFP, Rel -> SgpFP
      DeleteRelation(S, i) : SgpFP, RngIntElt -> SgpFP
      ReplaceRelation(S, r1, r2) : SgpFP, Rel, Rel -> SgpFP
      ReplaceRelation(S, i, r) : SgpFP, RngIntElt, Rel -> SgpFP
      AddGenerator(S) : SgpFP -> SgpFP
      AddGenerator(S, w) : SgpFP, SgpFPElt -> SgpFP
      DeleteGenerator(S, y) : SgpFP, SgpFPElt -> SgpFP

 
String Operations on Words
      Eliminate(u, x, v) : SgpFPElt, SgpFPElt, SgpFPElt -> SgpFPElt
      Match(u, v, f) : SgpFPElt, SgpFPElt, RngIntElt -> BoolElt, RngIntElt
      Random(S, m, n) : SgpFP, RngIntElt, RngIntElt -> SgpFPElt
      RotateWord(u, n) : SgpFPElt, RngIntElt -> SgpFPElt
      Substitute(u, f, n, v) : SgpFPElt, RngIntElt, SgpFPElt, RngIntElt -> SgpFPElt
      Subword(u, f, n) : SgpFPElt, RngIntElt, RngIntElt -> SgpFPElt
      ElementToSequence(u) : SgpFPElt -> [ SgpFPElt ]

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013