We are comparing the observation of a combined system or the combination of observations.
The other direction, restricting an observation of a system to a subsystem, does not have problems for monotone maps (which preserve meets, not joins).
A monotone map \(P \xrightarrow{f} Q\) that preserves meets
\(f(a \land_P b) \cong f(a) \land_Q f(b)\)
Likewise, to preserve joins is for \(f(a \lor_P b) \cong f(a) \lor_Q f(b)\)
A monotone map \(P \xrightarrow{f} Q\) has a generative effect
\(\exists a,b \in P: f(a) \lor f(b) \not\cong f(a \lor v)\)
Prove that for any monotone map \(P \xrightarrow{f} Q\):
if there exist \(a \lor b \in P\) and \(f(a) \lor f(b) \in Q\):
\(f(a) \lor_Q f(b) \leq f(a \lor_P b)\)
Let’s abbreviate \(f(a\ \lor_P\ b)\) as \(JF\) (join-first) and \(f(b)\ \lor_Q\ f(a)\) as \(JL\) (join-last)
This exercise is to show that \(JL \leq JF\)
The property of joins gives us, in \(P\), that \(a\ \leq\ (a \lor b)\) and \(b\ \leq\ (a \lor b)\)
Monotonicity then gives us, in \(Q\), that \(f(a) \leq JF\) and \(f(b) \leq JF\)
We also know from the property of joins, in \(Q\), that \(f(a) \leq JL\) and \(f(b) \leq JL\)
The only way that \(JF\) could be strictly smaller than \(JL\), given that both are \(\geq f(a)\) and \(\geq f(b)\) is for \(f(a) \leq JF < JL\) and \(f(b) \leq JF < JL\)
But, \(JL \in Q\) is the smallest thing (or equal to it) that is greater than \(f(a)\) and \(f(b)\), so this situation is not possible.