Modal group theory: homomorphisms

TL;DR

This paper explores modal group theory for arbitrary homomorphisms, showing how modal language can express various group properties and linking the homomorphic modal theory of finitely presented groups to true arithmetic.

Contribution

It introduces a novel interpretation of modal logic in the context of group homomorphisms and establishes a connection between group theory and arithmetic via definability results.

Findings

Modal language expresses subgroup properties like cyclicity and torsion.

Homomorphic modal theory of finitely presented groups is computably isomorphic to true arithmetic.

Sentential validities are exactly S5, and certain groups have validities S4.2.

Abstract

I investigate modal group theory for arbitrary homomorphisms. Possibility is interpreted by the existence of a group homomorphism out of the given group, so the semantics is governed by the possibility of collapse: elements may be identified, parameters may be killed, and new relations may hold in the target. I show that the modal language nevertheless expresses cyclic subgroup membership, subgroup generation by a fixed finite tuple, cyclicity, finite generation by a fixed number of elements, and torsion. I use these definability results to interpret arithmetic, and prove that, as sets of Goedel numbers, the homomorphic modal theory of finitely presented groups is computably isomorphic to true arithmetic. I also analyze propositional modal validities: sentential validities are exactly S5, the trivial group has exact parameter-validities S5, and uniformly prime-indivisible groups have exact parameter-validities S4.2.