Complete coinductive theories. I
A. H. Lachlan · Transactions of the American Mathematical Society · 1990
Let T T be a complete theory over a relational language which has an axiomatization by ∃ ∀ \exists \forall -sentences. The properties of models of T T are studied. It is shown that quantifier-free formulas are stable. This limited stability is used to show that in ∃ ∀ \exists \forall -saturated models the elementary types of tuples are determined by their ∃ \exists -types and algebraicity is determined by existential formulas. As an application, under the additional assumption that no quantifier-free formula has the FCP, the models M \mathcal {M} of T T are completely characterized in terms of certain 0 0 -definable equivalence relations on cartesian powers of M M . This characterization yields a result similar to that of Schmerl for the case in which T T is ℵ 0 {\aleph _0} -categorical.