连续格的谓词理论
Predicative theories of continuous lattices
AI总结：
本文提出强邻近并半格作为连续格的谓词概念，通过下幂格子和双幂格子的余代数刻画，并给出基于蕴涵关系的逻辑刻画，使点无拓扑中连续格的谓词与有限性质更明确。
摘要：
我们引入强邻近并半格的概念，这是一种连续格的谓词概念，作为代数格范畴的 Karoubi 包而出现。强邻近并半格可以通过在更广泛的邻近偏序集（也称为抽象基或 R-结构）范畴上的下幂格子的余代数来刻画。此外，局部紧 locale 可以通过在邻近偏序集范畴上的双幂格子的余代数，用强邻近并半格来刻画。我们还提供了强邻近并半格的更逻辑化的刻画，称为强连续有限覆盖，它使用蕴涵关系来呈现底层的并半格。我们表明该结构自然地对应于谓词点无拓扑中的连续格概念。我们的结果使点无拓扑中连续格概念的谓词和有限方面更加明确。
英文摘要：
We introduce a notion of strong proximity join-semilattice, a predicative notion of continuous lattice which arises as the Karoubi envelop of the category of algebraic lattices. Strong proximity join-semilattices can be characterised by the coalgebras of the lower powerlocale on the wider category of proximity posets (also known as abstract bases or R-structures). Moreover, locally compact locales can be characterised in terms of strong proximity join-semilattices by the coalgebras of the double powerlocale on the category of proximity posets. We also provide more logical characterisation of a strong proximity join-semilattice, called a strong continuous finitary cover, which uses an entailment relation to present the underlying join-semilattice. We show that this structure naturally corresponds to the notion of continuous lattice in the predicative point-free topology. Our result makes the predicative and finitary aspect of the notion of continuous lattice in point-free topology more explicit.
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