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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.01819 |
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Table of Contents:
- 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.