I was reading the wikipédia article about Interior algebra: https://en.wikipedia.org/wiki/Interior_algebra
And I found a passage that made me very curious:
"The open elements of an interior algebra form a Heyting algebra and the closed elements form a dual Heyting algebra. The regular open elements and regular closed elements correspond to the pseudo-complemented elements and dual pseudo-complemented elements of these algebras respectively and thus form Boolean algebras. The clopen elements correspond to the complemented elements and form a common subalgebra of these Boolean algebras as well as of the interior algebra itself. Every Heyting algebra can be represented as the open elements of an interior algebra and the latter may be chosen to an interior algebra generated by its open elements - such interior algebras correspond one to one with Heyting algebras (up to isomorphism) being the free Boolean extensions of the latter."
My question is: Where can I find a reference talking about this relation in more detail? I'm searching through the internet, but unfornutely I was not able yet to find a good book that exposes the topic in detail. Any help will be very useful
This is an example of a representation or duality principle. The ur-example of course is Stone duality/Stone's representation theorem, which is for Boolean (as opposed to Heyting) algebras; the standard text on this topic is Johnstone's wonderful book Stone spaces. I recall that it treats the result above as well, but I don't have my copy on hand right now; regardless, it's absolutely worth your time.
There's also Chagrov/Zakharyaschev's Modal Logic. This is a wonderful book, if quite advanced; interior algebras are studied there under the name "topological Boolean algebras." See especially chapters $7$ and $8$.
Finally, looking through my "stuff to read later" files I found Colin Naturman's Ph.D. thesis which seems extremely relevant; however, not having read it yet I can't actually vouch for it.