On maximal ladders

TL;DR

This paper investigates the size and existence of maximal n-ladders in lattice theory, using set-theoretic forcing and combinatorial principles to establish consistency results.

Contribution

It introduces the notion of maximal n-ladders and explores their cardinalities and existence under various set-theoretic assumptions.

Findings

Forcing with (, ___) makes all maximal n-ladders have size _{n-1}.

Existence of maximal 3-ladders of size _1 is consistent with certain set-theoretic axioms.

Under lubsuit, a maximal 3-ladder of breadth 2 can be constructed.

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 $\text{Add}(\omega, \omega_\omega)$ 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 $\clubsuit$, we construct a maximal $3$-ladder of breadth $2$. Finally, we prove that, consistently (under $\diamondsuit$), there exists a maximal $3$-ladder that is destructible by forcing with a Suslin tree.