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# resolutionPoset -- generates a poset from a resolution

- Usage:
    `P = resolutionPoset C`
    `P = resolutionPoset I`
- Inputs:
  - C, a complex,
  - I, a monomial ideal,
  - I, an ideal,
- Outputs:
  - P, an instance of the type Poset,

## Description

Given a resolution $C$, a poset can be defined by the non-zero entries of the matrices of each component of the resolution.

    i1 : R = QQ[x,y,z];

    i2 : C = freeResolution ideal(y*z,x*z,x^2*y)

          1      3      2
    o2 = R  <-- R  <-- R

         0      1      2

    o2 : Complex

    i3 : resolutionPoset C

    o3 = Relation Matrix: | 1 1 1 1 1 1 |
                          | 0 1 0 0 1 0 |
                          | 0 0 1 0 1 1 |
                          | 0 0 0 1 0 1 |
                          | 0 0 0 0 1 0 |
                          | 0 0 0 0 0 1 |

    o3 : Poset

    i4 : (resolutionPoset C).GroundSet

    o4 = {{0, 0}, {1, 0}, {1, 1}, {1, 2}, {2, 0}, {2, 1}}

    o4 : List

Moreover, the resolution-poset of a MonomialIdeal can be labeled as the lcm of the generators involved at each level. As the lcm needn't be unique at each step, we simply append it to the base labeling, as above.

    i5 : P = resolutionPoset monomialIdeal(y*z,x*z,x^2*y)

    o5 = P

    o5 : Poset

    i6 : P.GroundSet

                                  2                                      2
    o6 = {{0, 0, {0, 0}}, {1, 0, x y}, {1, 1, x*z}, {1, 2, y*z}, {2, 0, x y*z},
         ------------------------------------------------------------------------
         {2, 1, x*y*z}}

    o6 : List

## Ways to use resolutionPoset:

- `resolutionPoset(Complex)`
- `resolutionPoset(Ideal)`
- `resolutionPoset(MonomialIdeal)`

## For the programmer

The object resolutionPoset is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Posets.m2:3862:0.