Semitopologies model consensus in distributed system by equating the notion of a quorum -- a set of participants sufficient to make local progress -- with that of an open set. This yields a topology-like theory of consensus, but semitopologies generalise topologies, since the intersection of two quorums need not necessarily be a quorum. The semitopological model of consensus is naturally heterogeneous and local, just like topologies can be heterogenous and local, and for the same reasons: points may have different quorums and there is no restriction that open sets / quorums be uniformly generated (e.g. open sets can be something other than two-thirds majorities of the points in the space). Semiframes are an algebraic abstraction of semitopologies. They are to semitopologies as frames are to topologies. We give a notion of semifilter, which plays a role analogous to filters, and show how to build a semiframe out of the open sets of a semitopology, and a semitopology out of the semifilters of a semiframe. We define suitable notions of category and morphism and prove a categorical duality between (sober) semiframes and (spatial) semitopologies, and investigate well-behavedness properties on semitopologies and semiframes across the duality. Surprisingly, the structure of semiframes is not what one might initially expect just from looking at semitopologies, and the canonical structure required for the duality result -- a compatibility relation *, generalising sets intersection -- is also canonical for expressing well-behavedness properties. Overall, we deliver a new categorical, algebraic, abstract framework within which to study consensus on distributed systems, and which is also simply interesting to consider as a mathematical theory in its own right.

翻译：半拓扑通过将法定人数（足以实现局部进展的参与者集合）的概念等同于开集的概念，为分布式系统中的共识建模。这产生了一种类似拓扑的共识理论，但半拓扑推广了拓扑，因为两个法定人数的交集不一定必须是法定人数。共识的半拓扑模型本质上是异构和局部的，正如拓扑可以是异构和局部的一样，并且原因相同：点可以有不同的法定人数，并且没有限制要求开集/法定人数必须统一生成（例如，开集可以是空间中点的三分之二多数以外的其他形式）。半框架是半拓扑的一种代数抽象。它们之于半拓扑，犹如框架之于拓扑。我们给出了半滤子的概念，其作用类似于滤子，并展示了如何从半拓扑的开集构建半框架，以及如何从半框架的半滤子构建半拓扑。我们定义了合适的范畴和态射概念，并证明了（sober）半框架与（spatial）半拓扑之间的范畴对偶性，同时研究了对偶性下半拓扑和半框架的良好行为性质。令人惊讶的是，半框架的结构并非仅从观察半拓扑时最初可能预期的那样，并且对偶性结果所需的规范结构——一种推广集合交集的兼容关系*——对于表达良好行为性质也是规范的。总体而言，我们提供了一个新的范畴化、代数化、抽象的框架，用于研究分布式系统中的共识，并且其本身作为数学理论也颇具研究价值。