Connected Reduced Products

TL;DR

This paper characterizes when reduced products of binary structures are connected, linking set-theoretic cardinalities with properties of ultraproducts and filters, and establishing conditions under which connectivity is preserved.

Contribution

It provides a set-theoretic characterization of connectivity in reduced products, connecting large cardinal assumptions with model-theoretic properties.

Findings

Connectivity in reduced products relates to the size of the index set and set-theoretic assumptions.

Under certain conditions, ultraproducts of linear graphs are disconnected.

The results hold within ZFC for specific implications about connectivity.

Abstract

If $\rho$ is a binary relation on a set $X$, the structure ${\mathbb X}=\langle X,\rho\rangle$ is connected iff the minimal equivalence relation containing $\rho$ is the full relation on $X$. We show that, for a set $I$ the following conditions are equivalent (a) $|I|$ is less than the first measurable cardinal, (b) For each filter $\Phi \subset P(I)$ and each family $\{ {\mathbb X}_i :i\in I\}$ of binary structures, the reduced product $\prod_\Phi {\mathbb X}_i$ is connected, iff there are a finite set $K\subset I$ and $n\in \omega$ such that ${\mathbb X}_i$ is connected, for each $i\in K$, and $\{ i\in I: {\mathbb X}_i \mbox{ is of diameter }\leq n\}\cup K\in \Phi$, (c)The ultraproduct $\prod_{{\mathcal U}}{\mathbb G}_{\omega}$ is a disconnected graph for each non-principal ultrafilter ${\mathcal U}\subset P(I)$, where ${\mathbb G}_{\omega}$ is the linear graph on $\omega$. Moreover, the implication "$\Leftarrow$" in (b) holds in ZFC.