Near-discreteness of modules and spaces as measured by Gabriel and Cantor
type: Item
instance of: scholarly article
title: Near-discreteness of modules and spaces as measured by Gabriel and Cantor
zbMATH Open document ID: 0663.06007
DOI: 10.1016/0022-4049(89)90141-2
author: Harold Simmons
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: This paper erects a systematic theory to hold the diverse versions of Cantor-Bendixson analysis studied by the author in Lect. Notes Math. 915, 310-323 (1982; Zbl 0495.06005) and in Bull Lond. Math. Soc. 20, No.1, 16- 22 (1988; Zbl 0607.16021). The basic object is a modular meet-continuous lattice, here called an ``idiom'': a ``non-distributive frame''. Sober topological spaces are of course faithfully (contravariantly) represented in idiom theory. A module enters, non-faithfully, represented by its lattice of submodules. The theme is the comparison of extended Cantor-Bendixson analysis and Gabriel dimension. A general notion of a derivative f: \(\Lambda\) \(\to \Lambda\) on an idiom \(\Lambda\) is defined, covering the Cantor-Bendixson (CB) derivative and (in a suitable sense) the Gabriel and socle derivatives. The latter are defined, though, on the associated frame \(N\Lambda\), the lattice of congruence relations with order reversed. Naturally \(N\Lambda\) has its own CB derivative. \(\Lambda\) is said to have Gabriel rank if the idempotent closure of the Gabriel derivative is constant (in effect, it wipes \(\Lambda\) out). Idioms having Gabriel rank are characterized, generalizing \textit{J. Raynaud} [Publ. Dépt. Math., Lyon 8, No.3/4, 77-112 (1971; Zbl 0269.16003)]. For them, these three derivatives on \(N\Lambda\) coincide. Concluding the paper, the author reviews frames with Gabriel rank and the Gabriel dimension of a module and suggests some unsolved problems.
Mathematics Subject Classification ID: 06B35
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16B50
Mathematics Subject Classification ID: 06B23
zbMATH DE Number: 4083692
zbMATH Keywords: Cantor-Bendixson analysis
zbMATH Keywords: modular meet-continuous lattice
zbMATH Keywords: idiom
zbMATH Keywords: non- distributive frame
zbMATH Keywords: lattice of submodules
zbMATH Keywords: Gabriel dimension
zbMATH Keywords: derivative
zbMATH Keywords: lattice of congruence relations
zbMATH Keywords: Gabriel rank
reviewed by: J. R. Isbell
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90141-2
OpenAlex ID: W2058487089
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cites work: Q3804856
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cites work: Krull dimension
cites work: The Gabriel dimension of a module
cites work: Q4165427
cites work: Atomless Parts of Spaces.
cites work: Q4104974
cites work: Q3965241
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cites work: Spatial sublocales and essential primes
cites work: Q3215398
cites work: Q5686165
cites work: Q4180496
cites work: Q5536689
cites work: Spaces with Boolean assemblies
cites work: Q3958510
cites work: Torsion theoretic points and spaces
cites work: The Gabriel Dimension and Cantor-Bendixson Rank of a Ring
cites work: Q4049223
description: scientific article; zbMATH DE number 4083692