Canonical forests in directed families

TL;DR

This paper establishes uniqueness and minimality properties of set decompositions in directed families, leading to a one-dimensional elimination of imaginaries in certain model-theoretic theories.

Contribution

It introduces new uniqueness results for set representations in unpackable and packable cases within directed families, advancing understanding in VC-minimal theories.

Findings

Swiss cheese decompositions are unique in unpackable cases.

Minimal decompositions are unique under a quasi-order in packable cases.

Results enable one-dimensional elimination of imaginaries in VC-minimal theories.

Abstract

Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they are not unique but admit a quasi-ordering under which the minimal decomposition is unique. Both cases lead to a one-dimensional elimination of imaginaries in VC-minimal and quasi-VC-minimal theories.