Matrix iterations with vertical support restrictions

TL;DR

This paper introduces a new matrix iteration technique with vertical support restrictions to preserve unbounded and mad families in definable structures, broadening the understanding of forcing preservation properties.

Contribution

It develops the concept of $\sigma$-Frechet-linked posets and demonstrates their effectiveness in preserving mad and unbounded families during FS iterations.

Findings

Preservation of mad families added by Hechler's poset regardless of cofinality.

Introduction of $\sigma$-Frechet-linked posets for preservation results.

Consistency results for Cichoń's diagram with 7 values, including singular ones.

Abstract

We use coherent systems of FS iterations on a power set, which can be seen as matrix iteration that allows restriction on arbitrary subsets of the vertical component, to prove general theorems about preservation of certain type of unbounded families on definable structures and of certain mad families (like those added by Hechler's poset for adding an a.d. family) regardless of the cofinality of their size. In particular, we define a class of posets called $\sigma$-Frechet-linked and show that they work well to preserve mad families, and unbounded families on $\omega^\omega$. As applications of this method, we show that a large class of FS iterations can preserve the mad family added by Hechler's poset (regardless of the cofinality of its size), and the consistency of a constellation of Cicho\'n's diagram with 7 values where two of these values are singular.