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