Let $A\subseteq B\subseteq C$ are join-semilattices. (The order of $A$ is induced by the order of $B$, the order of $B$ is induced by the order of $C$.)
What is the best way to concisely formulate the conjunction of the following items:
- $x\cup^A y = x\cup^B y$ for every $x,y\in A$ (or equivalently $x\cup^B y\in A$).
- $x\cup^B y = x\cup^C y$ for every $x,y\in B$ (or equivalently $x\cup^C y\in B$).
We can also infer $x\cup^A y = x\cup^C y$ for every $x,y\in A$. Should this be formulated explicitly?