TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 169, July 1972 A SHEAF-THEORETIC DUALITY THEORY FOR CYLINDRIC ALGEBRAS BY STEPHEN D. COMER(') ABSTRACT. Stone's duality between Boolean algebras and Boolean spaces is extended to a dual equivalence between the category of all ^dimensional cylindric algebras and a certain category of sheaves of such algebras. The dual spaces of important types of algebras are characterized and applications are given to the study of direct and subdirect decompositions of cylindric algebras. It is a thesis of this paper that certain sheaves serve adequately as the dual spaces of cylindric algebras in the same way that Boolean spaces serve as the dual spaces of Boolean algebras. This duality is described in §1. These results are established by algebraically imitating, with suitable cylindric algebra concepts, the sheaf duality theory for rings presented in R. S. Pierce's mono- graph [6]. These results also hold for other versions of algebraic logic such as polyadic algebras. In ^2 the dual spaces of locally finite, representable, and regular algebras are characterized; §4 gives some applications to the decompo- sition theory for cylindric algebras. Our study can be viewed in several ways. In algebraic logic, with each first- order theory Y there is associated an algebraic structure (called an algebra of formulas) that describes certain aspects of Y. Since the theory Y can be deter- mined from the set of all complete theories extending Y, the following problem concerning the adequacy of algebraic logic arises. Assuming we know the alge- bra $£ associated with each complete (and consistent) theory A extending the theory Y, how can we describe the algebra %r associated with Y in terms of all the pairs (A, ^a)? This problem is similar to the one in algebraic geometry of de- scribing the ring associated with an affine variety in terms of the local rings given at each point of the variety. In our situation, if we think of a theory Y as being determined by the set Xp of all complete extensions of Y and think of the algebra of formula ^¡A as being assigned to each point A of Xp, then our problem is of the same nature as the one in algebraic geometry mentioned above. This analogue with algebraic geometry is very close; in §3 we solve the logical prob- Presented to the Society, April 5, 1969 under the title Representation of cylindric algebras by sheaves and January 23, 1970 under the title The dual space of an algebra of formulas; received by the editors October 14, 1970. AMS 1969 subject classifications. Primary 0240, 0242, 0248. Key words and phrases. Cylindric algebras, sheaves, sectional representations, Boolean spaces, Stone representation theorem, dual space of a cylindric algebra. (^Research supported by NSF Grants GP 8725 and GP 11804. — r Copyright © 1972. American Mathematical Society License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use