Finitely Presented Algebras

• 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
    • R ! s: AlgFr, RngElt → AlgFrElt
    • R ! s: AlgFr, [ RngElt ] → AlgFrElt
    • elt< R | s >: AlgFr, [ RngElt ] → AlgFrElt
    • One(F): AlgFr → AlgFrElt
    • Identity(F): AlgFr → AlgFrElt
    • Zero(F): AlgFr → AlgFrElt
    • Representative(F): AlgFr → AlgFrElt
• Structure Operations
  • Related Structures
    • BaseRing(F): AlgFr → Rng
    • CoefficientRing(F): AlgFr → Rng
    • Category(F): AlgFr → Cat
    • Parent(F): AlgFr → Pow
    • PrimeRing(F): AlgFr → Rng
  • Numerical Invariants
    • Rank(F): AlgFr → RngIntElt
    • Characteristic(F): AlgFr → RngIntElt
    • # F: AlgFr → RngIntElt
  • Homomorphisms
    • hom< F -> S | f, y₁, ..., yₙ >: AlgFr, Rng → Map
    • hom< F -> S | y₁, ..., yₙ >: AlgFr, Rng → Map
    • Example: Homomorphism
• Element Operations
  • Arithmetic Operators
    • + a: AlgFrElt → AlgFrElt
    • - a: AlgFrElt → AlgFrElt
    • a + b: AlgFrElt, AlgFrElt → AlgFrElt
    • a - b: AlgFrElt, AlgFrElt → AlgFrElt
    • a * b: AlgFrElt, AlgFrElt → AlgFrElt
    • a ^ k: AlgFrElt, RngIntElt → AlgFrElt
    • a / b: AlgFrElt, AlgFrElt → FldFunMElt
    • a div b: AlgFrElt, AlgFrElt → AlgFrElt
    • a +:= b: AlgFrElt, AlgFrElt → AlgFrElt
    • a -:= b: AlgFrElt, AlgFrElt → AlgFrElt
    • a *:= b: AlgFrElt, AlgFrElt → AlgFrElt
    • a div:= b: AlgFrElt, AlgFrElt → AlgFrElt
  • Equality and Membership
    • a eq b: AlgFrElt, AlgFrElt → BoolElt
    • a ne b: AlgFrElt, AlgFrElt → BoolElt
    • a in R: AlgFrElt, Rng → BoolElt
    • a notin R: AlgFrElt, Rng → BoolElt
  • Predicates on Algebra Elements
    • IsZero(f): AlgFrElt → BoolElt
    • IsOne(f): AlgFrElt → BoolElt
    • IsMinusOne(f): AlgFrElt → BoolElt
    • IsNilpotent(f): AlgFrElt → BoolElt
    • IsIdempotent(f): AlgFrElt → BoolElt
    • IsUnit(f): AlgFrElt → BoolElt
    • IsZeroDivisor(f): AlgFrElt → BoolElt
    • IsRegular(f): AlgFrElt → BoolElt
    • IsIrreducible(f): AlgFrElt → BoolElt
    • IsPrime(f): AlgFrElt → BoolElt
  • 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: Terms
  • Evaluation
    • Evaluate(f, s): AlgFrElt, [ RngElt ] → RngElt
    • Evaluate(f, s): AlgFrElt, < RngElt, ..., RngElt > → RngElt
    • Example: Terms
• Ideals and Gröbner Bases
  • Creation of Ideals
    • ideal<A | L>: AlgFr, List → AlgFr
    • lideal<A | L>: AlgFr, List → AlgFr
    • rideal<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: 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 ]
    • Reduce(S): { AlgFrElt } → [ AlgFrElt ]
    • Example: GB
• 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: Element Operations
• Changing Coefficient Ring
  • ChangeRing(I, S): AlgFr, Rng → AlgFr
• Finitely Presented Algebras
• Creation of FP-Algebras
  • quo< F | J >: AlgFr, AlgFr → AlgFP
  • quo< F | a₁, ..., aᵣ >: AlgFr, AlgFrElt, ..., AlgFrElt → AlgFrRes, Map
  • F / J: AlgFr, AlgFr → AlgFP
  • FPAlgebra< K, X | L >: Fld, List, List → AlgFP
  • Example: Creation
• 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: Finite Dimensional
• Vector Enumeration
  • Finitely Presented Modules
  • \(S\)-algebras
  • Finitely Presented Algebras
  • Vector Enumeration
    • (1) A permutation module
    • (2) A quotient of a permutation module
    • (3) A non-cyclic module
    • Example: Abstract
  • The Isomorphism
  • Sketch of the Algorithm
  • Weights
  • Setup Functions
    • FreeAlgebra(R, M): Rng, MonFP → AlgFPOld
    • FreeAlgebra(R, G): Rng, GrpFP → 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: Permutation Action D8
    • Example: Quotient