Residual stratification and the Cantor-Bendixson structures of dual algebraic coframes

TL;DR

This paper introduces residual derivatives for preordered sets, linking Cantor-Bendixson structures of dual algebraic coframes with residual derivatives, enabling a unified analysis across algebra, topology, and dynamics.

Contribution

It generalizes the Cantor-Bendixson derivative using residual derivatives and characterizes the first two levels in dual algebraic coframes, bridging multiple mathematical domains.

Findings

Established a partial correspondence between Cantor-Bendixson structures and residual derivatives.

Provided a complete characterization of the first two Cantor-Bendixson levels.

Unified the study of topological structures across algebra, analysis, and dynamics.

Abstract

We introduce a notion of residual derivative for elements of a preordered set, a construction that generalizes both the Frattini subgroup in algebra and the Cantor-Bendixson derivative in T1 topological spaces. For dual algebraic coframes with topologies compatible with order, we establish a partial correspondence between the Cantor-Bendixson structure of the lattice and the residual derivatives of its elements. Within this framework, we provide a complete characterization of the first two Cantor-Bendixson levels in terms of the lattice's residual structure. This provides a unified lens through which to study the Cantor-Bendixson structures of topological spaces across domains ranging from algebra to functional analysis and dynamics, facilitating the transfer of analytic techniques between them.