The Order-Sobrification Monad

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.