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 κ and λ are cardinal numbers, and let $$\mathcal F$$ be a $$\lambda$$-complete filter on $$\lambda$$. An Ulam matrix is a collection of subsets $$A_{\alpha \beta}$$ of $$\lambda$$ indexed by $$\alpha \in \kappa, \beta \in \lambda$$ such that
 * If $$\beta \ne \gamma \in \lambda$$ then $$A_{\alpha \beta}$$ and $$A_{\alpha \gamma}$$ are disjoint.
 * For each $$\beta \in \lambda$$, the union over $$\alpha \in \kappa$$ of the sets $$A_{\alpha \beta}, \, \bigcup\left\{A_{\alpha \beta}:\alpha \in \kappa\right\}$$, is in the filter $$\mathcal F$$.