Ulam matrix

Short description: Term in mathematical set theory

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 $\kappa$ and $\lambda$ are cardinal numbers, and let ${ℱ}$ 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 $A_{\alpha \beta },\bigcup \{A_{\alpha \beta }:\alpha \in \kappa \}$, is in the filter ${ℱ}$.

References

• Ulam, Stanisław (1930), "Zur Masstheorie in der allgemeinen Mengenlehre", Fundamenta Mathematicae 16 (1): 140–150, doi:10.4064/fm-16-1-140-150, http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-fmv16i1p14bwm?q=2c1badc7-22c8-4f89-a9de-dee5818f5258$8&qt=IN_PAGE