MonomialAlgebra -- The class of all monomialAlgebras.

Description

The class of monomial algebras K[B] where B is a subsemigroup of \mathbb{N}^r.

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 \mathbb{N}^2. 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:

affineAlgebra(MonomialAlgebra) -- see affineAlgebra -- Define a monomial algebra
binomialIdeal(MonomialAlgebra) -- see binomialIdeal -- Compute the ideal of a monomial algebra
codimMA(MonomialAlgebra) -- see codimMA -- Codimension of a monomial algebra.
decomposeHomogeneousMA(MonomialAlgebra) -- see decomposeHomogeneousMA -- Decomposition of one monomial algebra over a subalgebra
decomposeMonomialAlgebra(MonomialAlgebra) -- see decomposeMonomialAlgebra -- Decomposition of one monomial algebra over a subalgebra
degreeMA(MonomialAlgebra) -- see degreeMA -- Degree of a monomial algebra.
degrees(MonomialAlgebra) -- Generators of the degree monoid
findMonomialSubalgebra(MonomialAlgebra) -- see findMonomialSubalgebra -- Find monomial subalgebra corresponding to the convex hull.
isBuchsbaumMA(MonomialAlgebra) -- see isBuchsbaumMA -- Test whether a simplicial monomial algebra is Buchsbaum.
isCohenMacaulayMA(MonomialAlgebra) -- see isCohenMacaulayMA -- Test whether a simplicial monomial algebra is Cohen-Macaulay.
isGorensteinMA(MonomialAlgebra) -- see isGorensteinMA -- Test whether a simplicial monomial algebra is Gorenstein.
isNormalMA(MonomialAlgebra) -- see isNormalMA -- Test whether a simplicial monomial algebra is normal.
isSeminormalMA(MonomialAlgebra) -- see isSeminormalMA -- Test whether a simplicial monomial algebra is seminormal.
isSimplicialMA(MonomialAlgebra) -- see isSimplicialMA -- Test whether a monomial algebra is simplicial.
net(MonomialAlgebra) -- Pretty print for monomial algebras
regularityMA(MonomialAlgebra) -- see regularityMA -- Compute regularity from decomposition
ring(MonomialAlgebra) -- Multigraded polynomial ring associated to a monomial algebra

For the programmer

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

The source of this document is in MonomialAlgebras.m2:1234:0.