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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 = Poset{cache => CacheTable{}                                                                                                                                                                                                                }
           GroundSet => {{1, a}, {1, b}, {1, c}, {2, a}, {2, b}, {2, c}, {3, a}, {3, b}, {3, c}}
           RelationMatrix => | 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 |
           Relations => {{{1, a}, {2, a}}, {{1, b}, {2, b}}, {{1, c}, {2, c}}, {{2, a}, {3, a}}, {{2, b}, {3, b}}, {{2, c}, {3, c}}, {{1, a}, {1, b}}, {{2, a}, {2, b}}, {{3, a}, {3, b}}, {{1, b}, {1, c}}, {{2, b}, {2, c}}, {{3, b}, {3, c}}}

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 5;
i3 : B == product(5, i -> chain 2)

o3 = true
i4 : B == divisorPoset (2*3*5*7*11)

o4 = true
i5 : B == divisorPoset (2^2*3*5*7)

o5 = false

See also