When Locality Meets Preservation

TL;DR

This paper explores the expressiveness of a fragment of first-order logic related to locality and preservation properties, extending classical theorems and analyzing finite structures and decision problems.

Contribution

It establishes a new preservation theorem linking local sentences with local elementary embeddings, extending the L"os-Tarski Theorem.

Findings

Positive Boolean combinations of basic local sentences match sentences preserved under local elementary embeddings.

The full preservation theorem does not hold in finite structures and related decision problems are undecidable.

Preservation under extensions in finite structures is equivalent to local preservation.

Abstract

This paper investigates the expressiveness of a fragment of first-order sentences in Gaifman normal form, namely the positive Boolean combinations of basic local sentences. We show that they match exactly the first-order sentences preserved under local elementary embeddings, thus providing a new general preservation theorem and extending the L\'os-Tarski Theorem. This full preservation result fails as usual in the finite, and we show furthermore that the naturally related decision problems are undecidable. In the more restricted case of preservation under extensions, it nevertheless yields new well-behaved classes of finite structures: we show that preservation under extensions holds if and only if it holds locally.