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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:

  1. $x\cup^A y = x\cup^B y$ for every $x,y\in A$ (or equivalently $x\cup^B y\in A$).
  2. $x\cup^B y = x\cup^C y$ 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?

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I would say that the join operations in $A$ and $B$ agree with the join operation in $C$.

Often people use $\vee$ to denote the join operation for lattices.

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