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# Properties of the Lattice of Theories

(@jwrigley)
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My purpose in starting this thread is to garner what is understood about the lattice of theories $\mathfrak{Th}_\Sigma^{\mathbb{T}}$, for a given geometric theory $\mathbb{T}$, that is the theories $\mathbb{T}' \supseteq \mathbb{T}$ with the order defined by:

$\mathbb{T}'' \leqslant \mathbb{T}' \iff \mathbb{T}'' \vDash \varphi \vdash_{\vec{x}} \psi, \ \forall \, \varphi \vdash_{\vec{x}} \psi \in \mathbb{T}' .$

I recently came across this in Olivia's Theories, Sites and Toposes (TST) where it is shown that $\mathfrak{Th}_\Sigma^{\mathbb{T}}$ 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 $\mathbb{T} \cup \{\, \top \vdash_{[]} \varphi\,\}$ are complemented in $\mathfrak{Th}_\Sigma^{\mathbb{T}}$ if there exists a sentence $\psi$ such that $\mathbb{T} \vDash \top \vdash_{[]} \varphi \lor \psi, \ \varphi \land \psi \vdash_{[]} \bot$ (i.e. $\varphi$ is complemented).

As always, let me know if there are any mistakes.  I'd like to know what else we can say about $\mathfrak{Th}_\Sigma^{\mathbb{T}}$ (and generally generate discussion).  Of course, just reading more TST would be sound advice as I've yet to finish it.

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(@jwrigley)
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Joined: 4 months ago
Posts: 10
Topic starter

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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