My purpose in starting this thread is to garner what is understood about the lattice of theories 
I recently came across this in Olivia's Theories, Sites and Toposes (TST) where it is shown that
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 ![\mathbb{T} \cup \{\, \top \vdash_{[]} \varphi\,\}](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BT%7D+%5Ccup+%5C%7B%5C%2C+%5Ctop+%5Cvdash_%7B%5B%5D%7D+%5Cvarphi%5C%2C%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002)
![\mathbb{T} \vDash \top \vdash_{[]} \varphi \lor \psi, \ \varphi \land \psi \vdash_{[]} \bot](https://s0.wp.com/latex.php?latex=+%5Cmathbb%7BT%7D+%5CvDash+%5Ctop+%5Cvdash_%7B%5B%5D%7D+%5Cvarphi+%5Clor+%5Cpsi%2C+%5C+%5Cvarphi+%5Cland+%5Cpsi+%5Cvdash_%7B%5B%5D%7D+%5Cbot&bg=ffffff&fg=333333&s=0&c=20201002)
As always, let me know if there are any mistakes. I'd like to know what else we can say about
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.
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