A Combinatorial Version of the Svenonius Theorem on Definability

TL;DR

This paper presents a combinatorial version of the Svenonius theorem, characterizing definability in structures through permutations that almost preserve relations, avoiding logical notions like elementary equivalence.

Contribution

It introduces a new combinatorial approach to the Svenonius theorem using permutations of sequences, focusing solely on the original structure without logical concepts.

Findings

Relation definability characterized by permutations almost preserving structure relations

Permutation-based criterion for definability in structures

Avoids reliance on elementary equivalence or logical notions

Abstract

The Svenonius theorem describes the (first-order) definability in a structure in terms of permutations preserving the relations of elementary extensions of the structure. In the present paper we prove a version of this theorem using permutations of sequences over the original structure (these are permutations of sequences of tuples of the structure elements as well). We say that such a permutation $\varphi$ almost preserves a relation if for every sequence of its arguments the value of the relation on an $n$-th element of the sequence and on its image under $\varphi$ coincide for almost all numbers $n.$ We prove that a relation is definable in a structure iff the relation is almost preserved by all permutations almost preserving the relations of the structure. This version limits consideration to the original structure only and does not refer to any logical notion, such as "elementary equivalence".