Notions of density that imply representability in algebraic logic (Q1295366)

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 were formulated in a monograph of Henkin, Monk and Tarski [\textit{L. Henkin}, \textit{J. D. Monk} and \textit{A. Tarski}, Cylindric algebras. Part II (Stud. Log. Found. Math. 115, North-Holland, Amsterdam) (1985; Zbl 0576.03043)]. The authors 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. Then they show that every rectangularly dense cylindric algebra is representable, and extend this result to other classes of algebras of logic, for example quasi-polyadic algebras and substitution-cylindrification algebras with and without equality, relation algebras, and special Boolean monoids (the results of the monograph mentioned above are special cases of the authors' general theorems). Henkin and Tarski also introduced the notion of a rich cylindric algebra and proved in op. cit. that every rich cylindric algebra of finite dimension (or, more generally, of locally finite dimension) satisfying certain special identities, is representable. The authors introduce a modification of the notion of a rich algebra that renders it more naturally. In particular, under this modification richness becomes a density notion. Moreover, this notion of richness applies not only to algebras with equality, such as cylindric algebras, but also to algebras without equality. It is proved that a finite-dimensional algebra is rich iff 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 finite-dimensional (or locally finite-dimensional) rich algebra of logic is representable. The authors do not 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 finite-dimensional rich quasi-polyadic algebras without equality.