Thursday, April 30, 2015

Comparisons between sets

We can produce certain comparisons between sets such as based upon the intersection size. This is what leads to the classes of independent, linear, and antidisjoint families of sets. The independent families have no intersection between elements. Linear families have no intersection larger then one. And the antidisjoint families are complementary to the independent families in the sense that no elements lack intersection. Pairs of sets can only either be comparable or incomparable which is what leads to chain families and sperner families. In chain families each pair of sets is comparable. In sperner families each pair of sets is incomparable. Laminar families are produced by the union of independence and comparability as each element is either comparable or it has empty intersection. In uniform families each element has the same size. Each of these comparisons between sets corresponds to a clique family. The dependence condition of antidisjoint families produces line graphs and the comparability condition produces comparability graphs and cocomparability graphs of subclass containment orders.

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