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.
The lattice of theories has several interesting properties that make it useful for studying the relationships between scientific theories. These properties include:

Partial ordering: The lattice of theories is a partially ordered set, which means that there is a welldefined relation between theories that reflects their degree of inclusion or compatibility. This partial ordering allows us to compare different theories and determine which ones are more general or more specific than others.

Supremum and Infimum: The lattice of theories has a unique maximum element called the supremum and a unique minimum element called the infimum. The supremum represents the most general theory that includes all other theories in the lattice, while the infimum represents the most specific theory that is included in all other theories.

Closure properties: The lattice of theories is closed under certain operations, such as taking unions and intersections of theories. This means that if we take the union or intersection of two theories, the result will always be another theory in the lattice.

Complementarity: The lattice of theories has a complementation operation that allows us to find the complement of a theory, which represents the set of phenomena that are not explained by that theory. This complementation operation is important for identifying the limitations and weaknesses of a theory.

Coherence: The lattice of theories is coherent, which means that it satisfies certain logical principles such as the distributive law, associative law, and commutative law. This coherence property ensures that the lattice of theories is a wellbehaved mathematical structure that can be studied using rigorous mathematical methods.
Overall, the lattice of theories is a useful tool for studying the relationships between scientific theories and understanding their strengths and weaknesses. By analyzing the properties of the lattice of theories, we can gain insights into the structure of scientific knowledge and the process of scientific inquiry.