Model theory of term algebras revisited

TL;DR

The article revisits the model theory of absolutely free term algebras and their completions, building on Maltsev's framework and adding an EF-game based proof of completeness. It shows that completions are parametrized by the indecomposable count $k\in\{0,1,\dots,\omega\}$, with the standard models $\mathcal{F}_k$ being rigid and atomic for $k\ge1$, and that all theories $\mathcal{T}_k$ are stable but not superstable, nfcp, weakly eliminates imaginaries, and are normal and $1$-based with trivial forking; in particular, no infinite group is interpretable. Model completeness fails for $k\ge1$, while $T_0$ is the model companion of the locally free algebra theory. The work also provides new stability-theoretic proofs and pedagogical context for locally free algebras as a concrete setting to study fundamental model-theoretic dividing lines.

Abstract

Building on work of Maltsev on locally free algebras in finite purely functional languages, we revisit the model theory of (absolutely free) term algebras and their completions. Maltsev's analysis yields a natural axiomatization together with quantifier elimination to positive Boolean combinations of special formulas, and shows that the complete extensions are parametrized exactly by the number $k\in\{0,1,\dots,ω\}$ of indecomposable elements; for $1\le k\leω$ the standard model is the free term algebra on $k$ generators. We give a new, quantifier-elimination--free proof of completeness using Ehrenfeucht--Fraïssé games, and we establish several further structural properties of the standard models and theories. In particular, for $1\le k\leω$ we prove first-order rigidity and atomicity of the standard model. For every $0\le k\leω$ we show that the corresponding theory does not have the finite cover property and weakly eliminates imaginaries. We also provide new proofs of stability-theoretic features previously obtained by Belegradek: the theories are stable but not superstable, normal (hence $1$-based), and have trivial forking; consequently, no infinite group is interpretable in any model. Finally, we analyze model completeness and show that $T_0$ is the model companion of the theory of locally free algebras, while the theories with $k\ge 1$ are not model complete.

Theorems & Definitions (74)

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