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posetWedgeProduct -- constructs the wedge product of several posets

Description

Given a list L of posets all with unique minimal elements, the function returns the wedge product of all the posets. The wedge product is defined as follows: Suppose that for $1\leq i\leq t$ we have posets $P_i$ with unique least element $\ell_i$. Their $\bf{wedge product}$ is the set: \[ P_1 \vee P_2 \vee \cdots \vee P_t = \left(\bigsqcup_{i=1}^t P_i \right)/ (\ell_1=\ell_2=\cdots =\ell_t), \] (meaning that we take the disjoint union of the sets $P_i$ in which we identify all the $\ell_i$ into one element) with the partial order $a \leq b$ if and only if $a\leq b$ in $P_i$ for some $i$.

This is the special case of posetFiberProduct where the domain of the map is a $1$-vertex poset.

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing
 
i2 : I = ideal(x^3, y^2)

             3   2
o2 = ideal (x , y )

o2 : Ideal of R
 
i3 : posetWedgeProduct {booleanLattice 3, chain 4, getPoset(R/I)}

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

o3 : Poset

See also

Ways to use posetWedgeProduct:

  • posetWedgeProduct(List)
  • posetWedgeProduct(Poset,Poset)

For the programmer

The object posetWedgeProduct is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/MacaulayPosets.m2:1624:0.