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arXiv · 2605.04851

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

Silvère Gangloff·Alonso Nuñez
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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.

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Silvère Gangloff, Alonso Nuñez. 2026-05-06. Residual stratification and the Cantor-Bendixson structures of dual algebraic coframes. https://arxiv.org/abs/2605.04851

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