# affineAlgebra -- Define a monomial algebra

## 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 .