On full cylindric set algebras.
Thomas A. Sudkamp · Notre Dame Journal of Formal Logic · 1979
By a full cylindric set algebra of dimension a, full CSA α , where a is an ordinal number, we mean a systemwhere U is a non-empty set, A is the power set of a U, 0 is the empty set, U, Π, and ~ are the set theoretic union, intersection and complement on A, and for all K, λ < a, C iκ is a unary operation on A and D^χ is a constant defined as follows:C K X = {y: yt aι U and for some xe X we have x λ = 3λχfor all λ Φ κ\ for every X e A, and Diκ\={y: ye a U and y κ = y^ (cf.1.1.5,[2]).In section 1 we give an axiom system for a subclass of cylindric algebras, which we call strong cylindric algebras, and show that $( is a strong CA α , a < ω, if, and only if, 51 is isomorphic to a full CSA α .In section 2 we restrict our attention to the theory of strong CA 2 and show that it is definitionally equivalent to the theory of a subclass of relation algebras axiomatized by McKinsey [3].The notation of [1] is used, and a familiarity with chapter 1 of that book is assumed.We begin by introducing a piece of notation which will prove to be convenient.Definition 1.1 If % is a CA α , a < ω, and i < a, then Definition 1.2 By a strong cylindric algebra of dimension a, where a is an ordinal number less than ω, we mean a structure % = (A, +, , -, 0, 1, c^,d κλ ) K)λ