Ulam matrix on omega-one ($(A_{\alpha ,n})$)

A family $(A_{\alpha ,n}{)}_{\alpha <{\omega }_1,n<\omega }$ whose row at $\alpha$ partitions $(\alpha ,{\omega }_1)$ and whose fixed-column cells are pairwise disjoint. Choose injections $e_{\beta }:\beta \to \omega$ and put $A_{\alpha ,n}=\{\beta >\alpha :e_{\beta }(\alpha )=n\}$. This gives a matrix used to split stationary sets into many stationary pieces.

There are ${\aleph }_1$ pairwise disjoint stationary subsets of the nonzero countable limit ordinals. In a Ulam matrix on omega-one, each row partitions a stationary tail into countably many cells, so some cell is stationary. One column contains stationary cells in uncountably many rows, and the column cells are disjoint. Unions indexed by different subsets of ${\omega }_1$ then differ on a stationary set.