Ulam matrix

In mathematical set theory, an Ulam matrix is an array of subsets of a cardinal number with certain properties. Ulam matrices were introduced by Stanislaw Ulam in his 1930 work on measurable cardinals: they may be used, for example, to show that a real-valued measurable cardinal is weakly inaccessible.

Definition

Suppose that and are cardinal numbers, and let be a -complete filter on. An Ulam matrix is a collection of subsets of indexed by such that

If then and are disjoint.
For each, the union over of the sets, is in the filter.