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Magma
Computer • algebra
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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, y
1
, ..., y
n
> : AlgFr, Rng -> Map
Example
AlgFP_Homomorphism (H89E1)
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 (H89E2)
Evaluation
Evaluate(f, s) : AlgFrElt, [ RngElt ] -> RngElt
Example
AlgFP_Terms (H89E3)
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 (H89E4)
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 (H89E5)
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 (H89E6)
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 (H89E7)
Vector Enumeration
Finitely Presented Modules
S-algebras
Finitely Presented Algebras
Vector Enumeration
Example
AlgFP_Abstract (H89E8)
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 (H89E9)
Example
AlgFP_Quotient (H89E10)
Bibliography
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V2.28, 28 February 2025