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# affineAlgebra -- Define a monomial algebra

- Usage:
    `affineAlgebra P`
    `affineAlgebra B`
- Inputs:
  - P, a polynomial ring, multigraded, with B = degrees R and K = coefficientRing R, or
  - B, a list, In the case B is specified, K is set via the Option CoefficientField.
- Optional inputs:
  - CoefficientField => ..., default value ZZ/101, Option to set the coefficient field.
- Outputs:
  - a ring,

## Description

Returns the monomial algebra K[B]=P/binomialIdeal(P) associated to the degree monoid of the polynomial ring P.

    i1 : kk=ZZ/101

    o1 = kk

    o1 : QuotientRing

    i2 : B = {{1,2},{3,0},{0,4},{0,5}}

    o2 = {{1, 2}, {3, 0}, {0, 4}, {0, 5}}

    o2 : List

    i3 : S = kk[x_0..x_3, Degrees=> B]

    o3 = S

    o3 : PolynomialRing

    i4 : affineAlgebra S

                                S
    o4 = ----------------------------------------------
           3        2   6    2 3     4    3 2   5    4
         (x x  - x x , x  - x x , x x  - x x , x  - x )
           0 2    1 3   0    1 2   1 2    0 3   2    3

    o4 : QuotientRing

    i5 : affineAlgebra B

                           kk[x ..x ]
                               0   3
    o5 = ----------------------------------------------
           3        2   6    2 3     4    3 2   5    4
         (x x  - x x , x  - x x , x x  - x x , x  - x )
           0 2    1 3   0    1 2   1 2    0 3   2    3

    o5 : QuotientRing

    i6 : M = monomialAlgebra B

    o6 = kk[x ..x ]
             0   3

    o6 : MonomialAlgebra generated by {{1, 2}, {3, 0}, {0, 4}, {0, 5}}

    i7 : affineAlgebra M

                           kk[x ..x ]
                               0   3
    o7 = ----------------------------------------------
           3        2   6    2 3     4    3 2   5    4
         (x x  - x x , x  - x x , x x  - x x , x  - x )
           0 2    1 3   0    1 2   1 2    0 3   2    3

    o7 : QuotientRing

## Ways to use affineAlgebra:

- `affineAlgebra(List)`
- `affineAlgebra(MonomialAlgebra)`
- `affineAlgebra(PolynomialRing)`

## For the programmer

The object affineAlgebra is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MonomialAlgebras.m2:1427:0.