This command allows for fast computation of LCM lattices, which are particularly useful in the study of resolutions of monomial ideals. Specifically the LCM lattice is the set of all lcms of subsets of the generators of the ideal, partially ordered by divisability.
i1 : S = QQ[a,b,c,d];
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i2 : I = ideal (b^2-a*d, a*d-b*c, c^2-b*d);
o2 : Ideal of S
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i3 : M = monomialIdeal (b^2, b*c, c^2);
o3 : MonomialIdeal of S
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i4 : L = lcmLattice (I);
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i5 : L.GroundSet
2 2 3 2 2 2 2 2
o5 = {b - a*d, - b*c + a*d, c - b*d, - b c + a*b d + a*b*c*d - a d , b c -
------------------------------------------------------------------------
3 2 2 3 2 2 2 3 3 4
b d - a*c d + a*b*d , - b*c + b c*d + a*c d - a*b*d , - b c + b c*d +
------------------------------------------------------------------------
2 2 3 3 2 2 2 2 2 2 2 3
a*b c d + a*b*c d - a*b d - a*b c*d - a c d + a b*d }
o5 : List
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i6 : L.RelationMatrix
o6 = | 1 0 0 1 1 0 1 |
| 0 1 0 1 0 1 1 |
| 0 0 1 0 1 1 1 |
| 0 0 0 1 0 0 1 |
| 0 0 0 0 1 0 1 |
| 0 0 0 0 0 1 1 |
| 0 0 0 0 0 0 1 |
7 7
o6 : Matrix ZZ <--- ZZ
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i7 : LM = lcmLattice (M);
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i8 : LM.GroundSet
2 2 2 2 2 2
o8 = {b , b*c, c , b c, b c , b*c }
o8 : List
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i9 : LM.RelationMatrix
o9 = | 1 0 0 1 1 0 |
| 0 1 0 1 1 1 |
| 0 0 1 0 1 1 |
| 0 0 0 1 1 0 |
| 0 0 0 0 1 0 |
| 0 0 0 0 1 1 |
6 6
o9 : Matrix ZZ <--- ZZ
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