分配逆半群与非交换Stone对偶
Distributive inverse semigroups and non-commutative Stone dualities
摘要：
我们把分配逆半群理论发展为无顶元的分配格的类似物，并证明了它们与具有单位谱空间的标准广群胚是对偶的，其中我们的谱空间不一定是紧的.证明了布尔逆半群可以刻画为每个素滤子都是超滤子的分布逆半群，并给出了Hausdorff性的拓扑刻画.本文将分片拓扑的概念推广到分配逆半群，证明了每个分配逆半群都有布尔化。作为这一结果的应用，我们给出了逆半群的Paterson泛群胚的新解释，并通过发展我们所谓的紧覆盖理论，我们还为Exel的紧群胚提供了概念基础。
英文摘要：
We develop the theory of distributive inverse semigroups as the analogue of distributive lattices without top element and prove that they are in a duality with those etale groupoids having a spectral space of identities, where our spectral spaces are not necessarily compact. We prove that Boolean inverse semigroups can be characterized as those distributive inverse semigroups in which every prime filter is an ultrafilter; we also provide a topological characterization in terms of Hausdorffness. We extend the notion of the patch topology to distributive inverse semigroups and prove that every distributive inverse semigroup has a Booleanization. As applications of this result, we give a new interpretation of Paterson's universal groupoid of an inverse semigroup and by developing the theory of what we call tight coverages, we also provide a conceptual foundation for Exel's tight groupoid.
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