More on expressibility of satisfiability in submodels and extensions

TL;DR

This paper investigates the expressibility of modal operators related to satisfiability in submodels and extensions within infinitary languages, highlighting syntactic criteria and differences between submodel and extension operators.

Contribution

It provides a syntactic criterion for expressibility in finitary predicate languages and demonstrates the expressibility of extension operators via universal theories.

Findings

Infinitary languages are closed under submodel operators in many cases.

In monadic signatures, submodel operators are closed within these languages.

Extension operators, while sometimes inexpressible by a single sentence, are always expressible by a universal theory in certain languages.

Abstract

We study expressibility in infinitary languages of the modal operators associated with satisfiability of sentences of these languages in submodels and extensions of models. We give a syntactic criterion for expressibility in finitary predicate languages, show that in many cases infinitary languages are closed under the operator associated with submodels, and that this is so in any language with a purely monadic signature. Finally, we prove that in finitary or strongly compact languages, the operator associated with extensions, though can be inexpressible by a single sentence, is always expressible by a universal theory, in striking contrast with the submodel case.