AI 中文总结
本文基于Townsend的局部ic Priestley对偶性，通过Heyting框架、Esakia幺位与Esakia框架，构建出完全构造性的局部ic Esakia对偶性。
AI 中文摘要
Esakia对偶性是Heyting代数与Esakia空间之间的对偶等价关系，但传统证明依赖素理想定理从谱恢复代数，该选择原理不具备构造有效性。本文基于Townsend的局部ic Priestley对偶性，构建完全构造性的局部ic Esakia对偶性。代数侧采用新近提出的Heyting框架作为Heyting代数的无点版本；空间侧用两种无点替代物建模Esakia空间：其一，将Esakia幺位（locale）定义为Townsend引入的有序Stone幺位的子类，直接证明Townsend的等价关系可限制为Heyting框架与Esakia幺位之间的对偶；其二，利用锥形框架理论（其中框架上保并闭算子建模局部ic预序），将Esakia框架引入为Esakia空间的框架论类似物，证明Esakia框架等价于Heyting框架且对偶等价于Esakia幺位，且该等价关系可分解Townsend等价关系的限制，从而得到完全构造性的局部ic Esakia对偶性。
英文摘要
Esakia duality is the dual equivalence between Heyting algebras and Esakia spaces. However, the traditional proof uses the Prime Ideal Theorem to recover the algebra from its spectrum, a choice principle that is not constructively valid. We build on Townsend's localic Priestley duality to describe a fully constructive, localic Esakia duality. On the algebraic side, we use the recently introduced Heyting frames as the point-free version of Heyting algebras. On the spatial side, Esakia spaces are modeled by two point-free alternatives. First, Esakia locales are defined as a subclass of the ordered Stone locales introduced by Townsend, and it is shown directly that Townsend's equivalence restricts to a duality between Heyting frames and Esakia locales. Second, using the theory of conic frames, in which join-preserving closure operators on frames model localic preorders, we introduce Esakia frames as a frame-theoretic analogue of Esakia spaces. It is shown that Esakia frames are equivalent to Heyting frames and dually equivalent to Esakia locales, and that this factorises the restriction of Townsend's equivalence. This yields a fully constructive, localic Esakia duality.
Comments54 pages
