On thin-complete ideals of subsets of groups

TL;DR

This paper characterizes the structure of the smallest thin-complete family containing a given family of subsets in a group, revealing its complexity and properties, especially for countable non-torsion groups.

Contribution

It introduces the concept of thin-completion of a family of subsets in groups and analyzes its structure and complexity, including the case of finite subsets in countable non-torsion groups.

Findings

The thin-completion of an ideal in a group is itself an ideal.

For countable non-torsion groups, the thin-completion of finite subsets is coanalytic but not Borel.

The paper provides a structural description of thin-complete families in groups.

Abstract

Given a family $F$ of subsets of a group $G$ we describe the structure of its thin-completion $\tau^*(F)$, which is the smallest thin-complete family that contains $I$. A family $F$ of subsets of $G$ is called thin-complete if each $F$-thin subset of $G$ belongs to $F$. A subset $A$ of $G$ is called $F$-thin if for any distinct points $x,y$ of $G$ the intersection $xA\cap yA$ belongs to the family $F$. We prove that the thin-completion of an ideal in an ideal. If $G$ is a countable non-torsion group, then the thin-completion $\tau^*(F_G)$ of the ideal $F_G$ of finite subsets of $G$ is coanalytic but not Borel in the power-set $P_G$ of $G$.