Base matrices of various heights

TL;DR

This paper demonstrates the existence of base matrices of various heights, including the continuum c, under certain set-theoretic assumptions, extending previous results and answering open questions.

Contribution

It establishes the existence of base matrices of height c and other regular uncountable heights in Cohen and random models, generalizing classical theorems.

Findings

Existence of base matrices of height c when the continuum c is regular.

Existence of base matrices of any regular uncountable height ≤ c in Cohen and random models.

Answers to open questions by Fischer, Koelbing, and Wohofsky.

Abstract

A classical theorem of Balcar, Pelant, and Simon says that there is a base matrix of height h, where h is the distributivity number of P(omega)/fin. We show that if the continuum c is regular, then there is a base matrix of height c, and that there are base matrices of any regular uncountable height less or equal than c in the Cohen and random models. This answers questions of Fischer, Koelbing, and Wohofsky.