Complete Coinductive Theories. I

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Let T be a complete theory over a relational language which has an axiomatization by 3V-sentences.The properties of models of T are studied.It is shown that quantifier-free formulas are stable.This limited stability is used to show that in 3V-saturated models the elementary types of tuples are determined by their 3-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 L of T are completely characterized in terms of certain O-definable equivalence relations on cartesian powers of M. This characterization yields a result similar to that of Schmerl for the case in whichThis is the first of two papers about complete first-order theories which have axiomatizations consisting entirely of 3V-sentences over a relational language.We call theories with such axiomatizations 3V-theories.Schmerl has called them coinductive.This paper is a sequel to [7] which characterized complete theories over a relational language which have axiomatizations consisting of 3-sentences and V-sentences.(From now on the restriction to relational languages which obtains throughout will be tacit.)Such theories are called primitive.Following Hodkinson and Macpherson [5] a structure L is said to be finitely partitioned if there exists a partition M = FUX 1 U ... UXn of the universe of L invariant underand the pointwise stabilizer of M\X i in Aut(L) induces Sym(X i ).In [7] it was shown that a complete theory is primitive if and only if all its models are finitely partitioned.Some acquaintance with [7] would be very helpful to readers of the present work.Schmerl [10] studied No-categorical 3V-theories over a finite relational language and proved that the unique countable model of such a theory is cellular in the following sense.L is cellular if there exist a finite subset X of M and equivalence relations E, F on M\X such that(1) (M\X)jE is finite.(2) If C is an E-class and D an F -class, then IC n DI = 1.