On maximal ladders

Abstract

Given a positive integer $n$, an $n$-ladder is a lower finite lattice whose elements have at most $n$ lower covers. In 1984, Ditor proved that every $n$-ladder has cardinality at most ${\aleph }_{n-1}$ and asked whether this bound is sharp, i.e., whether for each $n$ there is an $n$-ladder of cardinality ${\aleph }_{n-1}$. We isolate the notion of maximal $n$-ladder and use it to study Ditor's problem and related questions. We show that $\text{Add}(\ddot{I}‰,\ddot{I}{‰}_{\ddot{I}}‰)$ forces every maximal $n$-ladder to have cardinality ${\aleph }_{n-1}$, and hence forces a positive answer to Ditor's question for every $n$. In particular, it is consistent that there are no maximal $3$-ladders of cardinality ${\aleph }_1$. However, we show that the existence of such a ladder follows from $\mathfrak{d}={\aleph }_1$. Under $♣$, we construct a maximal $3$-ladder of breadth $2$. Finally, we prove that, consistently (under $♢$), there exists a maximal $3$-ladder that is destructible by forcing with a Suslin tree.

39 pages