How does the logic of Set^P vary with the properties of P?

[Neil Barton <barton.neil.alexander@gmail.com>]

Dear All,

I am very suspicious the answer to this (family of) question(s) is
well-known, but I couldn't find anything after a bit of searching so
I'll ask anyway.

(I've also tried asking on MathOverflow, if anyone is interested:
https://mathoverflow.net/questions/378167/how-do-properties-of-a-partial-order-mathbbp-affect-the-logic-of-the-functo)

I am interested in how the logic associated with the algebra of
subobjects in the functor category Set^P (for a partial order P)
varies with different properties of P. Thus far, all I've been able to
find is:

Fact 1. P is (weakly) linearly-ordered iff the logic of the topos is
intuitionistic logic with the classical tautology (phi rightarrow psi)
vee (psi rightarrow phi) added (otherwise known as Dummett's Logic).

Fact 2. If P has a least element then the topos is disjunctive (i.e.
if y:1 to Omega and z:1 to Omega are truth-values, then y cup z = true
iff y = true or z = true). I *think* this implication can be reversed,
but I'm not sure.

I was wondering if anything more is known about how the logic of the
topos varies according to the properties of P (and vice versa)? I'd be
interested in any information here, but to make things more concrete,
is it known:

Q1. If the logic is affected when P is directed or has incompatible elements?

Q2. If P has incompatible elements, does the size of the largest
antichain matter?

Q3. What if P doesn't have a least element? (In particular can Fact
2's implication be reversed?)

Q4. P has (or doesn't have) a maximal element?

(An aside: In the presentation I'm most familiar with (namely
Goldblatt's book) there is a restriction that P be a small category. I
don't know whether this is essential for the results, or just made for
metamathematical ease/queasiness of dealing with a functor category
that can't be represented as anything small.)

Thanks for any pointers.

Best Wishes,

Neil


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