Characterize Powersets among CPOs Powersets can be seen as complete atomic boolean algebras. > Is it possible to characterize **complete atomic boolean algebras** (CABA) among **complete partial orders** (CPO)? For example, it is possible to characterize **completeHeyting Algebras** as **complete posets** where $$a \wedge \bigvee a_i = \bigvee a \wedge a_i.$$ In this sense a complete boolen algebra is just a complete partial order with an exactness property, the _infinitary distributivity law_. My guess, in fact my hope, is that CABAs correspond to **complete atomic regular posets**. By regular I mean that the poset is a regular category.

As indicated in this nlab page,

> Powersets are precisely atomic CPOs.

The definition of atom and atomic is a bit different from the one expected by lattice theorists. Traslating from mine terminology to theirs, _atomic_ means that **there is a join-dense subset of completely join-irreducible elements**.