My purpose in starting this thread is to garner what is understood about the lattice of theories , for a given geometric theory , that is the theories with the order defined by:

I recently came across this in Olivia's Theories, Sites and Toposes (TST) where it is shown that is a complete Heyting algebra. I'm wondering what other properties we are able to deduce.

I've attached some notes giving a sufficient condition for the open subtopoi to be complemented in terms of their corresponding theories, namely the theories of the form are complemented in if there exists a sentence such that (i.e. is complemented).

As always, let me know if there are any mistakes. I'd like to know what else we can say about (and generally generate discussion). Of course, just reading more TST would be sound advice as I've yet to finish it.

Indeed, had I read further in TST, I would have come across the very result I proved in the attachment above in Subsection 4.2.2.4, therein described as "not hard to prove using logical arguments". Hopefully my proof is testament to that.