一句话：用幺半伴随统一扩展 Stone/Priestley 对偶，并建立 residuation 代数与 profinite 序幺半群等新对偶。
本文研究如何统一并推广代数逻辑与代数语言理论中的扩展 Stone 型对偶。作者提出用幺半伴随扩展抽象范畴对偶，使经典扩展 Stone 与 Priestley 对偶的多种形式成为特例。作为应用，论文首先考察 residuation 代数：在格上加入 residual 算子以代数化语言导数；证明 derivation 代数的子范畴对偶等价于 profinite 序幺半群范畴，并限制出 Boolean residuation 代数与 profinite 幺半群的对偶。该对偶进一步细化以刻画 profinite 序幺半群的关系态射，其对偶为 residuation 代数的自然态射。其次，将扩展对偶用于集合与完全原子 Boolean 代数的离散情形，给出小范畴范畴之对偶的具体描述。
Extensions of Stone-type dualities have a long history in algebraic logic and have also been instrumental in proving results in algebraic language theory. We show how to extend abstract categorical dualities via monoidal adjunctions, subsuming various incarnations of classical extended Stone and Priestley duality as special cases, and providing the foundation for two new concrete dualities: First, we investigate residuation algebras, which are lattices with additional residual operators modeling language derivatives algebraically. We show that the subcategory of derivation algebras is dually equivalent to the category of profinite ordered monoids, restricting to a duality between Boolean residuation algebras and profinite monoids. We further refine this duality to capture relational morphisms of profinite ordered monoids, which dualize to natural morphisms of residuation algebras. Second, we apply the categorical extended duality to the discrete setting of sets and complete atomic Boolean algebras to obtain a concrete description for the dual of the category of all small categories.
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@article{LenkeUM25,
title = {Extended Stone Duality via Monoidal Adjunctions},
author = {Fabian Lenke and Henning Urbat and Stefan Milius},
journal = {Logical Methods in Computer Science},
volume = {Volume 21, Issue 4},
year = {2025},
doi = {10.46298/lmcs-21(4:4)2025}
}