posetClosedProduct -- constructs the closed product of several posets

Usage:

posetClosedProduct L

posetClosedProduct(P, Q)
Inputs:
- L, a list, a list of posets
- P, an instance of the type Poset,
- Q, an instance of the type Poset,
Outputs:
- an instance of the type Poset,

Description

Given a list of posets that all have a unique minimal and maximal element, the function returns the closed product of all the posets. The closed product is defined as follows: Suppose that for $1 \leq i \leq t$ we have posets $P_i$ with unique least element $\ell_i$ and unique largest element $L_i$. Their $\bf{closed product}$ is the set: \[ P_1 \diamond P_2 \diamond \cdots \diamond P_t = \left( \bigsqcup_{i=1}^t P_i \right) / (\ell_1 = \ell_2 = \cdots = \ell_t, L_1 = L_2 = \cdots = L_t ), \] (meaning that we take the disjoint union of the sets $P_i$ in which we identify all the $\ell_i$ into one element and all the $L_i$ elements 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 posetConnectedSum where the domain of the map is a $1$-vertex poset.

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing

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

             2   2
o2 = ideal (x , y )

o2 : Ideal of R

i3 : posetClosedProduct(chain 3, getPoset(R/I))

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

o3 : Poset

Ways to use posetClosedProduct:

posetClosedProduct(List)
posetClosedProduct(Poset,Poset)

For the programmer

The object posetClosedProduct 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.

posetClosedProduct -- constructs the closed product of several posets

Description

See also

Ways to use posetClosedProduct:

For the programmer