Finitely Presented Algebras
- Introduction
- Representation and Monomial Orders
- Exterior Algebras
- Creation of Free Algebras and Elements
- Structure Operations
- Element Operations
- Arithmetic Operators
+ a: AlgFrElt → AlgFrElt- a: AlgFrElt → AlgFrElta + b: AlgFrElt, AlgFrElt → AlgFrElta - b: AlgFrElt, AlgFrElt → AlgFrElta * b: AlgFrElt, AlgFrElt → AlgFrElta ^ k: AlgFrElt, RngIntElt → AlgFrElta / b: AlgFrElt, AlgFrElt → FldFunMElta div b: AlgFrElt, AlgFrElt → AlgFrElta +:= b: AlgFrElt, AlgFrElt → AlgFrElta -:= b: AlgFrElt, AlgFrElt → AlgFrElta *:= b: AlgFrElt, AlgFrElt → AlgFrElta div:= b: AlgFrElt, AlgFrElt → AlgFrElt
- Equality and Membership
- Predicates on Algebra Elements
IsZero(f): AlgFrElt → BoolEltIsOne(f): AlgFrElt → BoolEltIsMinusOne(f): AlgFrElt → BoolEltIsNilpotent(f): AlgFrElt → BoolEltIsIdempotent(f): AlgFrElt → BoolEltIsUnit(f): AlgFrElt → BoolEltIsZeroDivisor(f): AlgFrElt → BoolEltIsRegular(f): AlgFrElt → BoolEltIsIrreducible(f): AlgFrElt → BoolEltIsPrime(f): AlgFrElt → BoolElt
- Coefficients, Monomials, Terms, and Degree
Coefficients(f): AlgFrElt → [ RngElt ]LeadingCoefficient(f): AlgFrElt → RngEltTrailingCoefficient(f): AlgFrElt → RngEltMonomialCoefficient(f, m): AlgFrElt, AlgFrElt → RngEltMonomials(f): AlgFrElt → [ AlgFrElt ]LeadingMonomial(f): AlgFrElt → AlgFrEltTerms(f): AlgFrElt → [ AlgFrElt ]LeadingTerm(f): AlgFrElt → AlgFrEltTrailingTerm(f): AlgFrElt → RngEltLength(m): AlgFrElt → RngIntEltm[i]: AlgFrElt, RngIntElt → AlgFrEltTotalDegree(f): AlgFrElt → RngIntEltLeadingTotalDegree(f): AlgFrElt → RngIntElt- Example: Monomials and Terms of an Element
- Evaluation
- Arithmetic Operators
- Ideals and Gröbner Bases
- Basic Operations on Ideals
- Changing Coefficient Ring
- Finitely Presented Algebras
- Creation of FP-Algebras
- Operations on FP-Algebras
A . i: AlgFP, RngIntElt → AlgFPEltCoefficientRing(A): AlgFP → RngRank(A): AlgFP → RngIntEltDivisorIdeal(I): AlgFP → AlgFrPreimageIdeal(I): AlgFP → AlgFrPreimageRing(A): AlgFP → AlgFrOriginalRing(A): AlgFP → RngIsCommutative(A): AlgFP → BoolEltI eq J: AlgFP, AlgFP → BoolEltI subset J: AlgFP, AlgFP → BoolEltI + J: AlgFP, AlgFP → AlgFPI * J: AlgFP, AlgFP → AlgFPIsProper(I): AlgFP → BoolEltIsZero(I): AlgFP → BoolElt
- Finite Dimensional FP-Algebras
Dimension(A): AlgFP → RngIntEltVectorSpace(A): AlgFP → ModTupFld, MapMatrixAlgebra(A): AlgFP → AlgMat, MapAlgebra(A): AlgFP → AlgAss, MapRepresentationMatrix(f): AlgFPElt → AlgMatEltIsUnit(f): AlgFPElt → BoolEltIsNilpotent(f): AlgFPElt → BoolElt, RngIntEltMinimalPolynomial(f): AlgFPElt → RngUPol- Example: Functions for a Finite-Dimensional FP-Algebra
- Vector Enumeration
- Bibliography