On heights of distributivity matrices

TL;DR

This paper constructs models demonstrating the existence of distributivity matrices of various heights relative to the continuum, highlighting their properties and preservation under certain conditions.

Contribution

It introduces a method to build models with distributivity matrices of arbitrary regular height, extending understanding of their structure and relation to cardinal characteristics.

Findings

Distributivity matrices of height larger than  are consistent.

Both  =  and  <  are achievable for these matrices.

The construction preserves -Canjarness during the process.

Abstract

We construct a model in which there exists a distributivity matrix of regular height $\lambda$ larger than $\mathfrak{h}$; both $\lambda = \mathfrak{c}$ and $\lambda < \mathfrak{c}$ are possible. A distributivity matrix is a refining system of mad families without common refinement. Of particular interest in our proof is the preservation of $\mathcal{B}$-Canjarness.