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
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 (H91E1)
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 (H91E2)
Evaluation
Evaluate(f, s) : AlgFrElt, [ RngElt ] -> RngElt
Example AlgFP_Terms (H91E3)
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 (H91E4)
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 (H91E5)
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 (H91E6)
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 (H91E7)
Vector Enumeration
Finitely Presented Modules
S-algebras
Finitely Presented Algebras
Vector Enumeration
Example AlgFP_Abstract (H91E8)
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 (H91E9)
Example AlgFP_Quotient (H91E10)
Bibliography
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