Let $A$ be a small allegory (like in Freyd and Scedrov book, or in the Elephant of Johnstone), does it always exists a tabular allegory $B$ and a fully faithfull representation of $A$ in $B$ ?
I am especially interested in the case where $A$ and $B$ are locally complete and distributive. (in this case, one can assume that $B$ is the systemic completion of a small allegory, and hence the question is equivalent to find a fullyfaithfull representation of $A$ in $Rel(\mathcal{T})$ the allegory of relation of a topos $\mathcal{T}$
Hence, another equivalent way to state my question is:
If $Q$ is an involutive unitale quantale, whose underlying poset is a frame and which satisfy the modularity law, then does there exists a topos $\mathcal{T}$ and an object $X \in \mathcal{T}$ such that $Q$ is the quantale of relation on $X$.
My guess is that it is a reasonable conjecture, but that one do not know the answer yet...
