Abstract
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.
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I would like to thank the anonymous referees for their careful reading of our submission and for the many helpful comments and suggestions which improve the paper greatly.
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Communicated by Martín Escardó.
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This research was partially supported by Labex DigiCosme (Project ANR-11-LABEX-0045-DIGICOSME) operated by ANR as part of the program “Investissement d’Avenir” Idex Paris-Saclay (ANR-11-IDEX-0003-02).
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Jia, X. The Order-Sobrification Monad. Appl Categor Struct 28, 845–852 (2020). https://doi.org/10.1007/s10485-020-09599-6
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DOI: https://doi.org/10.1007/s10485-020-09599-6