Free sequences in P({\omega})/fin

TL;DR

This paper studies maximal free sequences in the Boolean algebra of subsets of natural numbers modulo finite sets, focusing on their minimal cardinality and consistency results related to cardinal characteristics of the continuum.

Contribution

It establishes the consistency of certain equalities and inequalities among cardinal characteristics, specifically $rak{i} = rak{f} < rak{u}$, using Shelah's model, and offers a clear presentation of this construction.

Findings

Proves the consistency of $rak{i} = rak{f} < rak{u}$.

Provides structural insights into free sequences in $rak{P}( ats)/fin$.

Offers a streamlined presentation of Shelah's model for these characteristics.

Abstract

We investigate maximal free sequences in the Boolean algebra $\mathcal{P}(\omega)/\mathrm{fin}$, as defined by D. Monk. We provide some information on the general structure of these objects and we are particularly interested in the minimal cardinality of a free sequence, a cardinal characteristic of the continuum denoted $\mathfrak{f}$. Answering a question of Monk, we demonstrate the consistency of $\omega_1 = \mathfrak{i} = \mathfrak{f} < \mathfrak{u} = \omega_2$. In fact, this consistency is demonstrated in the model of S. Shelah for $\mathfrak{i} < \mathfrak{u}$. Our paper provides a streamlined and mostly self contained presentation of this construction.