A decidable expansion of $(\Gamma,+,F)$ with the independence property

TL;DR

This paper introduces a decidable expansion of a finitely generated module with an injective endomorphism, incorporating automatic subsets and extending previous definability results in model theory.

Contribution

It defines a new decidable expansion of $( ext{Gamma},+,F)$ that includes all automatic subsets, building on prior work on F-sets and automata in model theory.

Findings

The expansion is shown to be decidable.

All automatic subsets are definable within this expansion.

The framework extends previous results on F-sets and automata.

Abstract

Let $(\Gamma,+,F)$ be a finitely generated $\mathbb Z[F]$-module where $F$ is an injective endomorphism of the abelian group $\Gamma$. We restrict ourselves to a finite automa presentable subclass, introduced by J. Bell and R. Moosa in "F-sets and finite automata. J. Th\'eor. Nombres Bordeaux 31 (2019), no. 1, 101-130" and define an expansion containing the $\mathcal F$-sets defined by R. Moosa and T. Scanlon in "Am. J. Math. 126 (2004), no. 3, p. 473-522", where every automatic subset is definable.