可数完全海廷代数乘积上的斯科特拓扑
Scott topologies on products of countable complete Heyting algebras
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- School of Computer Information Engineering, Nanchang Institute of Technology（南昌工程学院计算机与信息工程学院）
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中文总结 AI 辅助
本文证明斯科特拓扑与可数表示框架的任意乘积相容且乘积空间是素谱的，并指出可数生成不足以保证该性质，同时给出阿廷空间与代数 dcpo 的关联及二分法定理。
中文摘要 AI 辅助
我们证明斯科特拓扑与可数表示框架的任意乘积可交换，并且每个这样的乘积都具有一个素谱斯科特空间。特别地，这适用于可数完全海廷代数的任意乘积。对于一族一致空间，其开集格的斯科特乘积相容性等价于其拓扑和的一致性和。可数生成并不充分：一个可数生成的空间框架可以具有非素谱的斯科特空间，并且其平方不满足乘积恒等式。我们还证明斯科特非素谱框架和空间框架的基数谱是向上封闭的，并且在连续统假设下，由所有不可数基数组成。最后，每个阿廷 $T_0$ 网络空间都是 $B$-空间；如果它也是 $d$-空间，则它承载一个代数 dcpo 的斯科特拓扑。因此，每个阿廷交连续 dcpo 都是代数的。这产生一个二分法定理：具有非素谱斯科特空间的 dcpo 必定不满足交连续性或阿廷性。
英文摘要
We prove that the Scott topology commutes with arbitrary products of countably presented frames and that every such product has a sober Scott space. In particular, this holds for arbitrary products of countable complete Heyting algebras. For a family of consonant spaces, Scott-product compatibility of their open-set lattices is equivalent to consonance of their topological sum. Countable generation does not suffice: a countably generated spatial frame can have a non-sober Scott space and fail the product identity for its square. We also show that the cardinal spectra of Scott non-sober frames and spatial frames are upward closed and, under the Continuum Hypothesis, consist of all uncountable cardinals. Finally, every Artinian $T_0$ web space is a $B$-space; if it is also a $d$-space, it carries the Scott topology of an algebraic dcpo. Consequently, every Artinian meet-continuous dcpo is algebraic. This yields a dichotomy theorem: a dcpo with a non-sober Scott space must fail meet continuity or Artinianity.
