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Posets :: product(Poset,Poset)

product(Poset,Poset) -- computes the product of two posets

Synopsis

Description

The Cartesian product of the posets P and Q is the new poset whose ground set is the Cartesian product of the ground sets of P and Q and with partial order given by (a,b) ≤(c,d) if and only if a ≤c and b ≤d.
i1 : product(chain 3, poset {{a,b},{b,c}})

o1 = Relation Matrix: | 1 1 1 1 1 1 1 1 1 |
                      | 0 1 1 0 1 1 0 1 1 |
                      | 0 0 1 0 0 1 0 0 1 |
                      | 0 0 0 1 1 1 1 1 1 |
                      | 0 0 0 0 1 1 0 1 1 |
                      | 0 0 0 0 0 1 0 0 1 |
                      | 0 0 0 0 0 0 1 1 1 |
                      | 0 0 0 0 0 0 0 1 1 |
                      | 0 0 0 0 0 0 0 0 1 |

o1 : Poset
The product of n chains of length 2 is isomorphic to the boolean lattice on n elements. These are also isomorphic to the divisor lattice on the product of n distinct primes.
i2 : B = booleanLattice 4;
i3 : B == product(4, i -> chain 2)

o3 = true

See also