"Observation preserves order but not join" in Applied Category Theory book

Question

I'm reading Applied Category Theory. In chapter 1, they describe "systems" that are undirected graphs. Two systems (with the same sets of vertices) can be "joined" (∨) by taking the union of their edges. They point out that after joining two systems, vertices that were not connected might have become connected. Here we consider only systems with vertices $v_1, v_2, v_3$.

Let $\phi(A)$ denote the Boolean observation of whether $v_1$ and $v_2$ are connected in system A. They say:

"[The] observation ($\phi$) fails to preserve the join operation."

At first this sounded backward to me. Isn't it that joining fails to preserve the (connectedness) observation? So I continued. They define an ordering on systems, and an ordering on Booleans, and point out that

$\phi$ preserves order but not join

What does it mean for $\phi$ to preserve order? By the usual interpretation, it means:

A ≤ B ⇒ Φ(A) ≤ Φ(B)

This makes sense because they've defined ≤ on systems (are all connections in A present in B?) and on Booleans (True > False). So, returning to whether observation preserves join, we should replace order (≤) with join (∨):

A ∨ B ⇒ Φ(A) ∨ Φ(B)

But what on earth could A ∨ B mean as a Boolean?

The text explains with an example of two systems (A and B) for which Φ(A) is false and Φ(B) is false yet Φ(A ∨ B) is true. This seems to suggest that the meaning might be:

Φ(A ∨ B) $\nRightarrow$ (or $\neq$?) Φ(A) ∨ Φ(B)

This works, but seems like the wrong interpretation for two reasons:

1. They haven't defined join on Booleans at this point in the book. Surely they wouldn't expect that we've predicted this operation.
2. It doesn't fit the same pattern as for order.

Alternatively, if this is the expression for "Φ fails to preserve ∨," then "Φ preserves ≤" seems like it ought to be:

Φ(A ≤ B) ⇒ (or =) Φ(A) ≤ Φ(B)

But this makes no sense either. So what is it that they mean?

Answer 1

When they say "the observation $\Phi$ fails to preserve the join operation, they mean that in general $\Phi(A \vee B) \neq \Phi(A) \vee \Phi(B)$.

It can be a little bit confusing because "$\Phi$ preserves joins" means that an equality ($\Phi(A \vee B) = \Phi(A) \vee \Phi(B)$) holds, while "$\Phi$ preserves order" means that an implication ($A \leq B \Rightarrow \Phi(A) \leq \Phi(B)$) is true, but this is the standard usage in mathematics.