Ladders and Squares

TL;DR

This paper investigates specific lattice structures with bounded breadth and size, providing positive existence results under the axiom of constructibility and certain combinatorial principles.

Contribution

It proves the existence of join-semilattices and lower-finite lattices with prescribed properties assuming $ ext{V}= ext{L}$ and $ox_ ext{kappa}$ principles.

Findings

Positive answers to Ditor's questions under constructibility.

Existence of lattices with specified breadth and size.

Results hold consistently with ZFC assuming combinatorial principles.

Abstract

In 1984, Ditor asked two questions: (1) For each $n\in\omega$ and infinite cardinal $\kappa$, is there a join-semilattice of breadth $n+1$ and cardinality $\kappa^{+n}$ whose principal ideals have cardinality $< \kappa$? (2) For each $n \in \omega$, 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 $\square_\kappa$ holds for enough $\kappa$'s.