Notions of Density That Imply Representability in Algebraic Logic

Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable (as a cylindric set algebra). This theorem and its analogues for quasi-polyadic algebras with and without equality are formulated in Henkin-Monk-Tarski 1985]. We introduce a natural and more general notion of rectangular density that can be applied to arbitrary cylindric and quasi-polyadic algebras, not just atomic ones. We then show that every rectangularly dense cylindric algebra is representable, and we extend this result to other classes of algebras of logic, for example quasi-polyadic algebras and substitution-cylindrication algebras with and without equality, relation algebras, and special Boolean monoids. The results of op. cit. mentioned above are special cases of our general theorems. We point out an error in the proof of the Henkin-Monk-Tarski representation theorem for atomic equality-free quasi-polyadic algebras with rectangular atoms. The error consists in the implicit assumption of a property that does not, in general, hold. We then give a correct proof of their theorem. Henkin and Tarski also introduced the notion of a rich cylindric algebra and proved in op. cit. that every rich cylindric algebra of nite dimension (or, more generally, of locally nite dimension) satisfying certain special identities is representable. We introduce a modiication of the notion of a rich algebra that, in our opinion, renders it more natural. In particular,under this modiicationrichness becomes a density notion. Moreover, our notion of richness applies not only to algebras with equality, such as cylindric algebras, but also to algebras without equality. We show that a nite dimensional algebra is rich ii it is rectangularly dense and quasi-atomic; moreover, each of these conditions is also equivalent to a very natural condition of point density. As a consequence, every nite dimensional (or locally nite dimensional)rich algebra of logic is representable. We do not have to assume the validity of any special identities to establish this representability. Not only does this give an improvement of the Henkin-Tarski representation theorem for rich cylindric algebras, it solves positively an open problem in op. cit. concerning the representability of nite dimensional rich quasi-polyadic algebras without equality. Boolean algebra is an abstract algebraic theory that allows us to study the laws and theorems of propositional logic using modern algebraic methods. There