Notions of density that imply representability in algebraic logic
type: Item
instance of: scholarly article
title: Notions of density that imply representability in algebraic logic
zbMATH Open document ID: 0930.03100
DOI: 10.1016/S0168-0072(98)00032-3
author: Steven R. Givant
author: Szabolcs Mikulás
author: András Simon
author: Hajnal Andréka
author: István Németi
published in: Annals of Pure and Applied Logic
publication date: 2000-02-15T00:00:00Z
review text: 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 establ…
Mathematics Subject Classification ID: 03G15
Mathematics Subject Classification ID: 03C05
Mathematics Subject Classification ID: 06E25
Mathematics Subject Classification ID: 08B05
zbMATH DE Number: 1308053
zbMATH Keywords: rectangular density
zbMATH Keywords: quasi-polyadic algebras
zbMATH Keywords: relation algebras
zbMATH Keywords: Boolean monoids
zbMATH Keywords: rich algebra
zbMATH Keywords: richness
zbMATH Keywords: cylindric algebras
zbMATH Keywords: finite-dimensional algebra
zbMATH Keywords: point density
zbMATH Keywords: representation theorem
zbMATH Keywords: representability
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
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full work available at URL: s0168-0072(98)00032-3
OpenAlex ID: W2014969671
description: scientific article; zbMATH DE number 1308053