Ladders and Squares

Abstract

In 1984, Ditor asked two questions: (1) For each $n\in \ddot{I}‰$ and infinite cardinal $\hat{I}º$, is there a join-semilattice of breadth $n+1$ and cardinality $\hat{I}{º}^{+n}$ whose principal ideals have cardinality $<\hat{I}º$? (2) For each $n\in \ddot{I}‰$, is there a lower-finite lattice of cardinality ${\aleph }_n$ whose elements have at most $n+1$ lower covers? We show that both questions have positive answers under the axiom of constructibility, and hence consistently with $\mathsf{ZFC}$. More specifically, we derive the positive answers from assuming that ${\square }_{\hat{I}}º$ holds for enough $\hat{I}º$'s.

30 pages