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MonomialAlgebras :: MonomialAlgebra

MonomialAlgebra -- The class of all monomialAlgebras.

Description

The class of monomial algebras K[B] where B is a subsemigroup of ℕr.

You can create a monomial algebra via the function monomialAlgebra by either specifying

- the semigroup B as a list of generators. The field K is selected via the option CoefficientField.

- a list of positive integers which is converted by adjoinPurePowers and homogenizeSemigroup into a list B of elements of ℕ2. The field K is selected via the option CoefficientField.

- a multigraded polynomial ring K[X] with Degrees R = B.

This data can be extracted as follows:

ring(MonomialAlgebra) returns the associated multigraded polynomial ring.

degrees(MonomialAlgebra) returns B.

Key functions:

Decomposition:

decomposeMonomialAlgebra -- Decomposition of a monomial algebra over the subalgebra corresponding to the convex hull of the degree monoid.

decomposeHomogeneousMA -- Decomposition of a homogeneous monomial algebra over the subalgebra corresponding to the convex hull of the degree monoid.

Ring-theoretic properties:

isCohenMacaulayMA -- Test whether a simplicial monomial algebra is Cohen-Macaulay.

isGorensteinMA -- Test whether a simplicial monomial algebra is Gorenstein.

isBuchsbaumMA -- Test whether a simplicial monomial algebra is Buchsbaum.

isNormalMA -- Test whether a simplicial monomial algebra is normal.

isSeminormalMA -- Test whether a simplicial monomial algebra is seminormal.

isSimplicialMA -- Test whether a monomial algebra is simplicial.

Regularity:

regularityMA -- Compute the regularity via the decomposition.

degreeMA -- Compute the degree via the decomposition.

codimMA -- Compute the codimension of a monomial algebra.

Methods that use a MonomialAlgebra :

For the programmer

The object MonomialAlgebra is a type, with ancestor classes MutableHashTable < HashTable < Thing.