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FINITELY PRESENTED ALGEBRAS

 
Acknowledgements
 
Introduction
 
Representation and Monomial Orders
 
Exterior Algebras
 
Creation of Free Algebras and Elements
      Creation of Free Algebras
      Print Names
      Creation of Polynomials
 
Structure Operations
      Related Structures
      Numerical Invariants
      Homomorphisms
 
Element Operations
      Arithmetic Operators
      Equality and Membership
      Predicates on Algebra Elements
      Coefficients, Monomials, Terms and Degree
      Evaluation
 
Ideals and Gröbner Bases
      Creation of Ideals
      Gröbner Bases
      Verbosity
      Related Functions
 
Basic Operations on Ideals
      Construction of New Ideals
      Ideal Predicates
      Operations on Elements of Ideals
 
Changing Coefficient Ring
 
Finitely Presented Algebras
 
Creation of FP-Algebras
 
Operations on FP-Algebras
 
Finite Dimensional FP- Algebras
 
Vector Enumeration
      Finitely Presented Modules
      S-algebras
      Finitely Presented Algebras
      Vector Enumeration
      The Isomorphism
      Sketch of the Algorithm
      Weights
      Setup Functions
      The Quotient Module Function
      Structuring Presentations
      Options and Controls
      Weights
      Limits
      Logging
      Miscellaneous
 
Bibliography







DETAILS

 
Introduction

 
Representation and Monomial Orders

 
Exterior Algebras

 
Creation of Free Algebras and Elements

      Creation of Free Algebras
            FreeAlgebra(K, n) : Fld, RngIntElt -> AlgFr
            ExteriorAlgebra(K, n) : Fld, RngIntElt -> AlgExt

      Print Names
            AssignNames(~F, s) : AlgFr, [ MonStgElt ]) ->
            Name(F, i) : AlgFr, RngIntElt -> AlgFrElt

      Creation of Polynomials
            F . i : AlgFr, RngInt -> AlgFrElt
            elt< R | a > : AlgFr, RngElt -> AlgFrElt

 
Structure Operations

      Related Structures
            BaseRing(F) : AlgFr -> Rng

      Numerical Invariants
            Rank(F) : AlgFr -> RngIntElt

      Homomorphisms
            hom< F -> S | f, y1, ..., yn > : AlgFr, Rng -> Map
            Example AlgFP_Homomorphism (H82E1)

 
Element Operations

      Arithmetic Operators

      Equality and Membership

      Predicates on Algebra Elements

      Coefficients, Monomials, Terms and Degree
            Coefficients(f) : AlgFrElt -> [ RngElt ]
            LeadingCoefficient(f) : AlgFrElt -> RngElt
            TrailingCoefficient(f) : AlgFrElt -> RngElt
            MonomialCoefficient(f, m) : AlgFrElt, AlgFrElt -> RngElt
            Monomials(f) : AlgFrElt -> [ AlgFrElt ]
            LeadingMonomial(f) : AlgFrElt -> AlgFrElt
            Terms(f) : AlgFrElt -> [ AlgFrElt ]
            LeadingTerm(f) : AlgFrElt -> AlgFrElt
            TrailingTerm(f) : AlgFrElt -> RngElt
            Length(m) : AlgFrElt -> RngIntElt
            m[i] : AlgFrElt, RngIntElt -> AlgFrElt
            TotalDegree(f) : AlgFrElt -> RngIntElt
            LeadingTotalDegree(f) : AlgFrElt -> RngIntElt
            Example AlgFP_Terms (H82E2)

      Evaluation
            Evaluate(f, s) : AlgFrElt, [ RngElt ] -> RngElt
            Example AlgFP_Terms (H82E3)

 
Ideals and Gröbner Bases

      Creation of Ideals
            ideal<A | L> : AlgFr, List -> AlgFr
            Basis(I) : AlgFr -> [ AlgFrElt ]
            BasisElement(I, i) : AlgFr, RngIntElt -> AlgFrElt

      Gröbner Bases
            Groebner(I: parameters) : AlgFr ->
            GroebnerBasis(I: parameters) : AlgFr -> AlgFrElt
            GroebnerBasis(S: parameters) : [ AlgFrElt ] -> [ AlgFrElt ]
            GroebnerBasis(S, d: parameters) : [ AlgFr ], RngInt -> AlgFrElt

      Verbosity
            SetVerbose("Groebner", v) : MonStgElt, RngIntElt ->
            SetVerbose("Buchberger", v) : MonStgElt, RngIntElt ->
            SetVerbose("Faugere", v) : MonStgElt, RngIntElt ->

      Related Functions
            MarkGroebner(I) : AlgFr ->
            Reduce(S) : [ AlgFrElt ] -> [ AlgFrElt ]
            Example AlgFP_GB (H82E4)

 
Basic Operations on Ideals

      Construction of New Ideals
            I + J : AlgFr, AlgFr -> AlgFr
            I * J : AlgFr, AlgFr -> AlgFr
            F / J : AlgFr, AlgFr -> AlgFrRes
            Generic(I) : AlgFr -> AlgFr

      Ideal Predicates
            I eq J : AlgFr, AlgFr -> BoolElt
            I ne J : AlgFr, AlgFr -> BoolElt
            I notsubset J : AlgFr, AlgFr -> BoolElt
            I subset J : AlgFr, AlgFr -> BoolElt
            IsZero(I) : AlgFr -> BoolElt

      Operations on Elements of Ideals
            f in I : AlgFrElt, AlgFr -> BoolElt
            NormalForm(f, I) : AlgFrElt, AlgFr -> AlgFrElt
            NormalForm(f, S) : AlgFrElt, [ AlgFrElt ] -> AlgFrElt
            f notin I : AlgFrElt, AlgFr -> BoolElt
            Example AlgFP_ElementOperations (H82E5)

 
Changing Coefficient Ring
      ChangeRing(I, S) : AlgFr, Rng -> AlgFr

 
Finitely Presented Algebras

 
Creation of FP-Algebras
      quo< F | J > : AlgFr, AlgFr -> AlgFP
      F / J : AlgFr, AlgFr -> AlgFP
      FPAlgebra< K, X | L > : Fld, List, List -> AlgFP
      Example AlgFP_Creation (H82E6)

 
Operations on FP-Algebras
      A . i : AlgFP, RngIntElt -> AlgFPElt
      CoefficientRing(A) : AlgFP -> Rng
      Rank(A) : AlgFP -> RngIntElt
      DivisorIdeal(I) : AlgFP -> AlgFr
      PreimageIdeal(I) : AlgFP -> AlgFr
      PreimageRing(A) : AlgFP -> AlgFr
      OriginalRing(A) : AlgFP -> Rng
      IsCommutative(A) : AlgFP -> BoolElt
      I eq J : AlgFP, AlgFP -> BoolElt
      I subset J : AlgFP, AlgFP -> BoolElt
      I + J : AlgFP, AlgFP -> AlgFP
      I * J : AlgFP, AlgFP -> AlgFP
      IsProper(I) : AlgFP -> BoolElt
      IsZero(I) : AlgFP -> BoolElt

 
Finite Dimensional FP- Algebras
      Dimension(A) : AlgFP -> RngIntElt
      VectorSpace(A) : AlgFP -> ModTupFld, Map
      MatrixAlgebra(A) : AlgFP -> AlgMat, Map
      Algebra(A) : AlgFP -> AlgAss, Map
      RepresentationMatrix(f) : AlgFPElt -> AlgMatElt
      IsUnit(f) : AlgFPElt -> BoolElt
      IsNilpotent(f) : AlgFPElt -> BoolElt, RngIntElt
      MinimalPolynomial(f) : AlgFPElt -> RngUPol
      Example AlgFP_FiniteDimensional (H82E7)

 
Vector Enumeration

      Finitely Presented Modules

      S-algebras

      Finitely Presented Algebras

      Vector Enumeration
            Example AlgFP_Abstract (H82E8)

      The Isomorphism

      Sketch of the Algorithm

      Weights

      Setup Functions
            FreeAlgebra(R, M) : Rng, MonFP -> AlgFPOld

      The Quotient Module Function
            QuotientModule(A, S) : AlgFPOld, AlgFPOld -> [AlgMatElt], [ModTupFldElt], [AlgFPEltOld]

      Structuring Presentations

      Options and Controls

      Weights
            QuotientModule(A, S) : AlgFP, AlgFP -> AlgFP

      Limits
            QuotientModule(A, S) : AlgFP, AlgFP -> AlgFP

      Logging
            QuotientModule(A, S) : AlgFP, AlgFP -> AlgFP

      Miscellaneous
            QuotientModule(A, S) : AlgFP, AlgFP -> AlgFP
            Example AlgFP_PermutationActionD8 (H82E9)
            Example AlgFP_Quotient (H82E10)

 
Bibliography

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