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# 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 /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MonomialAlgebras.m2:1234:0.