Upward Morley's theorem downward (Q2856635)

This paper studies the possibility to extend Morley's theorem to the finite case. Morley's theorem states that a countable first-order theory is \(\aleph_1\)-categorical if and only if it is \(\kappa\)-categorical for all uncountable \(\kappa\) (for a reference see [\textit{M. Morley}, Trans. Am. Math. Soc. 114, 514--538 (1965; Zbl 0151.01101)]). The authors show that \(\aleph_1\)-categorical theories not necessarily have a unique \(n\)-element model for each finite natural number, up to isomorphism. They present finitary analogues to the notions of elementary substructures and, in their main result (Theorem 1.2.) they show under which conditions a theory can have what they call the \textit{Finite Morley Property} (FMP). A theory \(T\) has the FMP if for every natural number \(n\), \(T\) has at most one \(n\)-element model, up to isomorphism. They introduce a new method of extending elementary maps between certain uncountably categorical structures. In the case of strongly minimal structures, they prove that the conditions on Theorem 1.2. can be simplified. Examples to which their theorem can be applied are infinite dimensional vector spaces over a finite field, and algebraically closed fields of a given positive characteristic.