Model theory of term algebras revisited

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 $k\in\{0,1,\dots,\omega\}$ of indecomposable elements; for $1\le k\le\omega$ the standard model is the free term algebra on $k$ generators. We give a new, quantifier-elimination--free proof of completeness using Ehrenfeucht--Fra\"iss\'e games, and we establish several further structural properties of the standard models and theories. In particular, for $1\le k\le\omega$ we prove first-order rigidity and atomicity of the standard model. For every $0\le k\le\omega$ 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.