# lcmLattice -- generates the lattice of lcms in an ideal

## Synopsis

• Usage:
P = lcmLattice I
• Inputs:
• Optional inputs:
• Strategy => an integer, default value recursive, either "subsets" or "recursive" (default)
• Outputs:
• P, an instance of the type Poset,

## Description

The LCM lattice of an Ideal is the set of all LCMs of subsets of the generators of the ideal with partial ordering given by divisibility. These are particularly useful in the study of resolutions of monomial ideals. Note that the minimal element of an LCM lattice will always be defined to be $1$ in the ring $R$ containing $I$ rather than $1$ in ZZ.

 i1 : R = QQ[x,y]; i2 : L = lcmLattice monomialIdeal(x^2, x*y, y^2) o2 = L o2 : Poset i3 : compare (L, 1_R, x^2*y);

Note that if $I$ is not a MonomialIdeal, then the method automatically uses the Strategy "subsets."

• lcm -- least common multiple

## Ways to use lcmLattice :

• "lcmLattice(Ideal)"

## For the programmer

The object lcmLattice is .