in the book by Katrin Tent and Martin Ziegler, "A course in model theory", it says that a complete theory in a finite relational language has quantifier eliminiation iff any isomorphism between finite substructures is elementary (Lemma 4.4.6.). I find this confusing since I thought that any isomorphism between two structures must be elementary, i. e. preserve the validity of all formulas. Can someone please find an error in my thought process and explain this? Thanks in advance.
2026-03-24 21:59:42.1774389582
Isomorphism between finite structures which is not elementary
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