Sets of natural numbers of positive density and cylindric set algebras of dimension 2
type: Item
instance of: scholarly article
title: Sets of natural numbers of positive density and cylindric set algebras of dimension 2
zbMATH Open document ID: 0473.03057
DOI: 10.1007/BF02483866
author: Jean A. Larson
author: Vance Faber
author: P. Erdős
published in: Algebra Universalis
publication date: 1981-01-01T00:00:00Z
review text: Among others, the reported paper solves some fundamental problems of Ulam concerning projective algebras. To this, the paper uses and improves the theory of cylindric set algebras and their generalizations. Since the reported paper investigates structures treated systematically and extensively in the textbook by \textit{L.Henkin, J.D.Monk, A.Tarski, H.Andréka} and \textit{I. Németi} Cylindric set algebras, Lect. Notes Math. 883 (1981; Zbl 0497.03025) on cylindric-relativized set algebras and their generalizations, we shall use the notation and terminology of this book. To this book we shall refer as HMTAN. The broadest generalization of cylindric set algebras of dimension \(\alpha(Cs_{\alpha}-s)\) is the class \(Crs_{\alpha}\) of cylindric-relativized set algebras, see Def. I.1.1 in HMTAN p.4. The reported paper introduces two special classes of \(Crs_{\alpha}-s\) which we shall denote by \(Crs_{\alpha}^{rc}\) and \(Crs_{\alpha}^{rcd}\). Definition: A \(Crs_{\alpha}\mathcal A\) is said to be rectangular if its unit \(1^{\mathcal A}\) is a Cartesian product, that is if \(1^{\mathcal A}=P_{i\in I}^{\alpha}W_i\) for some function \(W=\langle W_i:i\in I\rangle\). We let \(Crs_{\alpha}^{rc} := \{\mathcal A\in Crs_{\alpha}:\mathcal A\) is rectangular\}. A \(Crs_{\alpha}:\mathcal A\) is said to be disjointly rectangular if \(1^{\mathcal A}=P_{i\in I}W_i\) for some function \(\langle W_i:i\in I\rangle\) such that \((\forall i,j\in I)[i\neq j\Rightarrow W_i\cap W_j=0]\). We let \(Crs_{\alpha}^{rcd} := \{\mathcal A\in Crs_{\alpha}^{rc}:\mathcal A\) is disjointly rectangular\}. The two new classes introduced in the reported paper are \(Crs_{\alpha}^{rc}\) and \(Crs_{\alpha}^{rcd}\). The reported paper uses a terminology different from the one introduced here which we use in order to avoid confusion with the standard terminoloy of cylindric set algebra theory. The reported paper calls \(Crs_{\alpha}^{rc}-s\) ``cylindric set algebras with diagonal'' and \(Crs_{\alpha}^{rcd}-s\) …
Mathematics Subject Classification ID: 03G15
Mathematics Subject Classification ID: 05A99
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11U99
Mathematics Subject Classification ID: 03B10
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 08A02
zbMATH DE Number: 3741390
zbMATH Keywords: representable cylindric algebras
zbMATH Keywords: number of generators
zbMATH Keywords: algebras of relations
zbMATH Keywords: number of non-isomorphic algebras
zbMATH Keywords: projective algebras
Wikidata QID: Q105478744
MaRDI profile type: Publication
cites work: Generators for algebras of relations
cites work: Projective Algebra I
full work available at URL: https://doi.org/10.1007/bf02483866
OpenAlex ID: W1991236348
description: scientific article; zbMATH DE number 3741390