Scott拓扑在交连续域上的研究
Scott topologies on meet-continuous domains
- Nanchang Institute of Technology（南昌工程学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究可数交连续域的Scott积与sobriety性质，利用测试族定理证明有限积等式条件，并刻画sobriety，但一般情形仍开放。
AI中文摘要：
我们研究可数交连续域的Scott积和sobriety（sobriety，即每个不可约闭集都是某个点的闭包的性质），这里的交连续域指的是交连续的dcpo（有向完备偏序），且不要求额外的连续性或最小元假设。利用Xu和Ji的完全格测试族定理，我们证明了对于L-dcpo类以及更弱的、其主理想具有所有非空子集上确界的类，有限积等式成立。对于任意非空可数交连续L-dcpo族，我们证明序积上的Scott拓扑等于各因子Scott拓扑的乘积，当且仅当只有有限多个因子缺少最小元。我们还在有界完备性和额外的共同上界条件下建立了sobriety性质。假设平方积等式成立，sobriety可由与不可约Scott闭集相关的共同上界截口的Scott闭性来刻画，在可数情形下等价于序列形式。两个提取引理推广到交半格dcpo。一般可数交连续域的有限积和sobriety问题在此仍未解决。
英文摘要：
We study Scott products and sobriety of countable meet-continuous domains, meaning meet-continuous dcpos without any additional continuity or least-element assumption. Using the complete-lattice test-family theorem of Xu and Ji, we prove finite-product equality for those domains that are $L$-dcpos, and for the weaker class whose principal ideals have suprema of all nonempty subsets. For an arbitrary family of nonempty countable meet-continuous $L$-dcpos, we prove that the Scott topology on the order product equals the product of the factor Scott topologies if and only if only finitely many factors lack a least element. We also establish sobriety under bounded completeness and under additional common-upper-bound conditions. Assuming square-product equality, sobriety is characterized by Scott closedness of common-upper-bound sections associated with irreducible Scott-closed sets, with an equivalent sequential formulation in the countable case. Two extraction lemmas extend to meet-semilattice dcpos. The finite-product and sobriety questions for general countable meet-continuous domains remain unresolved here.
