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Preprint Number 135

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135. Domenico Zambella
Noetherianity in a class of first-order theories
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Submission date: 28 October 2008

Abstract:

We call a theory a Dedekind theory if every quantier-free type with one free variable either has a trivial positive part or it is isolated by a positive quantier-free formula. Vector spaces and integral domains are examples. We prove that in a Dedekind theory all positive quantier-free types are isolates so, in a sense, Dedekind theories are Noetherian. We show that saturated existentially closed models of Dedekind theories are Zariski geometries.

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