Complete coinductive theories. I

Lachlan, A. H.

doi:10.1090/S0002-9947-1990-0957082-6

Complete coinductive theories. I
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Trans. Amer. Math. Soc. 319 (1990), 209-241 Request permission

Abstract:

Let $T$ be a complete theory over a relational language which has an axiomatization by $\exists \forall$-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 $\exists \forall$-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 $\mathcal {M}$ of $T$ are completely characterized in terms of certain $0$-definable equivalence relations on cartesian powers of $M$. This characterization yields a result similar to that of Schmerl for the case in which $T$ is ${\aleph _0}$-categorical.

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Remarks on

-stable

-categorical theories

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