内邻域结构III：子空间的有限和
Internal Neighbourhood Structures III: Finite Sum of Subobjects
摘要：
在\cite{2020}中，提出了在具有有限余积和真$(\mathsf{E}, \mathsf{M})$系统的完备范畴上的内预邻域空间的概念，使得对于每个对象$X$，$X$的$\mathsf{M}$ -子范畴的集合是完全格.在\cite{2021-clos}中研究了闭包算子、闭态射及其近似关系的概念。本文给出了三元组$(\mathbb{A}, \mathsf{E}, \mathsf{M})$（$\mathbb{A}$外延）等价于有限和闭对象的$\mathsf{M}$ -子代数集的结构条件.给出了闭嵌入（闭态射）在有限和下闭的等价条件。当容许子格（闭嵌入）在有限和下闭时，证明了有限和的容许子格（闭嵌入）的联半格是其分支联半格的双积.最后，它表明，只要一套封闭态射是封闭的有限总和，一套适当的（分别，分离）态射也是封闭的有限总和。这导致了紧（分别为Hausdorff）预邻域空间的满子范畴在有限和下封闭的等价条件。
英文摘要：
The notion of an internal preneighbourhood space on a finitely complete category with finite coproducts and a proper $(\mathsf{E}, \mathsf{M})$ system such that for each object $X$ the set of $\mathsf{M}$-subobjects of $X$ is a complete lattice was initiated in \cite{2020}. The notion of a closure operator, closed morphism and its near allies investigated in \cite{2021-clos}. The present paper provides structural conditions on the triplet $(\mathbb{A}, \mathsf{E}, \mathsf{M})$ (with $\mathbb{A}$ lextensive) equivalent to the set of $\mathsf{M}$-subobjects of an object closed under finite sums. Equivalent conditions for the set of closed embeddings (closed morphisms) closed under finite sums is also provided. In case when lattices of admissible subobjects (respectively, closed embeddings) are closed under finite sums, the join semilattice of admissible subobjects (respectively, closed embeddings) of a finite sum is shown to be a biproduct of the component join semilattices. Finally, it is shown whenever the set of closed morphisms is closed under finite sums, the set of proper (respectively, separated) morphisms are also closed under finite sums. This leads to equivalent conditions for the full subcategory of compact (respectively, Hausdorff) preneighbourhood spaces to be closed under finite sums.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
