Residual stratification and the Cantor-Bendixson structures of dual algebraic coframes
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.
Cite
@article{arxiv.2605.04851,
title = {Residual stratification and the Cantor-Bendixson structures of dual algebraic coframes},
author = {Silvère Gangloff and Alonso Nuñez},
journal= {arXiv preprint arXiv:2605.04851},
year = {2026}
}