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I’ve inserted proofs of equivalence of the three conditions listed in the definition of De Morgan algebra.
I added the Sierpinski space as an example, though it looks like from De Morgan Topos any “extremally disconnected space” is an example.
Would be nice to have more concrete instances.
It appears that there are two different notions of “de Morgan algebra”. The one on our page De Morgan algebra is a Heyting algebra that satisfies de Morgan’s law; but the version on Wikipedia (which is also the one used in cubical type theory) is a distributive lattice equipped with a contravariant involution. Neither implies the other: in the former case the negation need not be involutive (otherwise it would be a Boolean algebra), while in the latter case the involution need not be induced from a Heyting implication.
What do we do?
We could have a disambiguation at the beginning, and link to a separate page Stone algebra for the thing we currently have (since Stone algebra is mentioned as an alternative term).
“Stone algebra” is used in the La Palme ReyesReyesZolfaghari-book and presumably by Gonzalo E. Reyes in general. “Stone lattice” has the authority of Birkhoff’s lattice theory book as well as Johnstone’s 1977 book in its favor.
On the other hand, the terminology as reported seems to be fairly well established in the topos theory literature and changing it would suggest changing De Morgan topos to Stone topos etc. (in view of things like “De Morganization” one might consider this an advantage though.)
I would think it better to suggest terminological preferences in a less definitive way, let’s say in a remark section. Then people working with the concepts can figure out what they think best.
So for the time being I would be inclined to keep both concepts under more and less the same name and on either the same or separate pages depending on how much attention&space one intends to give the involutive concept.
I don’t think it’s a good idea to have two different concepts on the same page.
The name “de Morgan topos” suggests a back-formation of “de Morgan locale”, and for the not-necessarily-complete version we could say “de Morgan Heyting algebra”.
The name de Morgan Heyting algebra occurred to me too.
No one objected to #6, so I went ahead and moved the current page to De Morgan Heyting algebra (and updated links) and created a new page De Morgan algebra for the other one. We can still discuss terminology if anyone wants, though; the page could be moved again if we reach a different consensus. I just wanted to resolve the issue in some way, so that for instance the link from cubical type theory would not be wrong.
Add the four-point algebra as an example. This is a great counterexample, and also generates all DeMorgan algebras (in Set) as subalgebras of products. Note that $\neg a = a, notb‘.
Didn’t include this fact at the moment as I didn’t have a citation handy, but it’s useful in that algebraic statements (with equality and regular logic) can be checked semantically against just this example.
Anonymous
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