(0,1)-category theory: logic, order theory
proset, partially ordered set (directed set, total order, linear order)
meet, logical conjunction, and
join, logical disjunction, or
lattice of subobjects
complete lattice, algebraic lattice
distributive lattice, completely distributive lattice, canonical extension
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A completely distributive lattice is a
in which arbitrary joins and arbitrary meets distribute over each other.
The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattices.
This appears as (Caramello, remark 4.3).
wikipedia, completely distributive lattice
Olivia Caramello, A topos-theoretic approach to Stone-type dualities (arXiv:1103.3493)