Near-discreteness of modules and spaces as measured by Gabriel and Cantor
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.
scientific article; zbMATH DE number 4083692
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