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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 complete Heyting 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.

Ivan Di Liberti
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2 Answers2

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Assuming by complete you mean in the categorical sense, i.e. you want limit-closure (i.e. meet-closure in the poset case), I suppose that your complete boolean algebras should be the same as complete star-autonomous categories where the monoidal product is the cartesian one.

In these categories you have an involution operator (the $*$ in star-autonomous) which plays the role of the negation and indeed you have, in the posetal case, de morgan laws and the middle excluded.

Edit: I see thatthe OP was looking for a property-like definition, hence the following addendum. As shown in the link above $*$-autonomous categories admit a definition in a almost-property like way (as long as we are considering just cartesian closed category) using the dualizing object $(\bot)$ instead of the involution functor $(*)$.

This is an almost property like, as opposed to property like, because we require the object $\bot$ to satisfy the condition

the morphism $A \to ((A \multimap \bot) \multimap \bot)$ to be an isomorphism

the problem is that in a general category this object would be an additional structure on the category.

But if our cartesian closed category, $*$-auotnomous, is a poset $\bot$ is necessarily the bottom element, i.e. the minimum.

To see that one can observe that if $ ((A \multimap \bot) \multimap \bot) \leq A$, which is the poset-version of the property above, since obviously $\bot \leq ((A \multimap \bot) \multimap \bot)$ it follows that $\bot \leq A$.

From this it follows that in a star-autonomous (cartesian) poset the global dualizing object must be the minimum of the poset.

So a star-autonomous-poset is nothing but a cartesian-closed-posetal category with a minimum $\bot$ that satisfies the property $$((A \multimap \bot) \multimap \bot \leq A$$

This should provide a property-like definition as you wished....or at least I hope so :-D

Giorgio Mossa
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    I am sorry to say that this is not the kind of answer I am looking for. In fact the $*$ operator is an additional structure, and not a property.

    In particular the operator $\neg$ is perfectly definable in a complete lattice as $\neg a := \bigvee_{d : d \wedge a = \bot} d$. Thus one could state the property of "having the $\neg$ operator" as an exactness property. What I really want to know is if this exactness property coincides with a very classical one, such as regularity, or being Heyting or whatever.

     – Ivan Di Liberti Aug 07 '18 at 17:54
  • @IvanDiLiberti I've made a little addendum that should address your concerns. Let me know if it's ok. – Giorgio Mossa Aug 07 '18 at 19:24
  • Thanks, this is perfect. – Ivan Di Liberti Aug 08 '18 at 05:33
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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.

Ivan Di Liberti
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