Sheaf constructions and their elementary properties

TLDR: In this article, a new definition of sheaf constructions over Boolean spaces is given, which is closely related to the formation of reduced products, in particular the universal Horn class generated by a class is the class of structures which can be embedded into the Boolean products of the given class.

Abstract: We are interested in sheaf constructions in model-theory, so an attempt is made to unify and generalize the results to date, namely various forms of the Feferman-Vaught Theorem, positive decidability results, and constructions of model companions. The task is considerably simplified by introducing a new definition of sheaf constructions over Boolean spaces. We start the paper by giving a new formulation in §1 of what sheaf constructions over Boolean spaces are-perhaps an appropriate name for our construction would be 'Boolean product'. In §2 we show that this construction is closely related to the formation of reduced products, in particular the universal Horn class generated by a class is the class of structures which can be embedded into the Boolean products of the given class. A specialized Boolean product operator F is indeed remarkably similar to reduced products, for in §3 it is shown that V preserves a sentence iff it is equivalent to a Horn sentence; and in §4 we take another look at Comer's version of the Feferman and Vaught Theorem (for sheaves) and show that it is the most general of those currently in use. Several model-theoretic results proved by Waszkiewicz and Weglorz for reduced limit powers have natural extensions to Boolean products as one also sees in §4. In §5 we take a construction introduced by Arens and Kaplansky, which we call a filtered Boolean power, and show how to extend a translation introduced by Ershov to such structures-this translation is a considerable simplification of the Feferman and Vaught approach of §4. Then in §6 we prove the result which motivated us to look at the elementary properties of sheaf constructions, namely that countable Boolean products of finitely many finite structures with a finite language have a decidable theory. This generalizes Comer's results for monadic algebras and (xm = x)-rings, and Ershov's results for bounded Boolean powers. The next section contains technical results needed for the study of model Received by the editors September 17, 1976 and, in revised form, February 8, 1977. AMS (MOS) subject classifications (1970). Primary 02H13, 02G05; Secondary 02G10, 02G15, 02G20, 08A20.