We investigate the so-called order-sobrification monad proposed by Ho et al. (Log Methods Comput Sci 14:1–19, 2018) for solving the Ho–Zhao problem, and show that this monad is commutative. We also show that the Eilenberg–Moore algebras of the order-sobrification monad over dcpo’s are precisely the strongly complete dcpo’s and the algebra homomorphisms are those Scott-continuous functions preserving suprema of irreducible subsets. As a corollary, we show that this monad gives rise to the free strongly complete dcpo construction over the category of posets and Scott-continuous functions. A question related to this monad is left open alongside our discussion, an affirmative answer to which might lead to a uniform way of constructing non-sober complete lattices.

中文翻译：

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Order-Sobrification Monad

我们研究了 Ho 等人提出的所谓的 order-sobrification monad。(Log Methods Comput Sci 14:1-19, 2018) 用于解决 Ho-Zhao 问题，并证明该 monad 是可交换的。我们还表明，在 dcpo 上的 order-sobrification monad 的 Eilenberg-Moore 代数恰好是强完备的 dcpo，而代数同态是那些保存不可约子集至上的 Scott 连续函数。作为推论，我们证明了这个 monad 在偏序集和 Scott 连续函数的类别上产生了自由的强完备 dcpo 构造。与这个 monad 相关的问题与我们的讨论一起悬而未决，一个肯定的答案可能会导致构建非清醒完整格的统一方法。