This is the monoid whose elements correspond to degrees of rings with heft vector x, or, in case x is an integer, of rings with degree rank x and no heft vector; see heft vectors. Hilbert series and polynomials of modules over such rings are elements of its monoid ring over ZZ; see hilbertPolynomial and hilbertSeries The monomial ordering is chosen so that the Hilbert series, which has an infinite number of terms, is bounded above by the weight.
i1 : degreesMonoid {1,2,5}
o1 = [T , T , T , Degrees => {1..2, 5}, MonomialOrder => {MonomialSize => 32 }, DegreeRank => 1, Inverses => true, Global => false]
0 1 2 {Weights => {-1, -2, -5}}
{GroupLex => 3 }
{Position => Up }
o1 : GeneralOrderedMonoid
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i2 : degreesMonoid 3
o2 = [T , T , T , Degrees => {3:{}}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 0, Inverses => true, Global => false]
0 1 2 {Weights => {3:-1} }
{GroupLex => 3 }
{Position => Up }
o2 : GeneralOrderedMonoid
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i3 : R = QQ[x,y,Degrees => {{1,-2},{2,-1}}];
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i4 : heft R
o4 = {1, 0}
o4 : List
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i5 : degreesMonoid R
o5 = [T , T , Degrees => {1, 0}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1, Inverses => true, Global => false]
0 1 {Weights => {-1..0}}
{GroupLex => 2 }
{Position => Up }
o5 : GeneralOrderedMonoid
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i6 : S = QQ[x,y,Degrees => {-2,1}];
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i7 : heft S |
i8 : degreesMonoid S^3
o8 = [T, Degrees => {{}}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 0, Inverses => true, Global => false]
{Weights => {-1} }
{GroupLex => 1 }
{Position => Up }
o8 : GeneralOrderedMonoid
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