Dan, your answer in #150 seems to suggest that \\(\emptyset\\) is a subset of some poset \\( A\\). So then if I were to assume that \\( A =\emptyset\\), then does the meet of \\(\emptyset\\) not exist?

I also want to understand what you meant by start as far up as we can when it comes to the empty set in #151. So going back to Alex's 2nd example in #147, it is true that \\(g_1\\) is not a right adjoint but it also does not preserve meet since:

$$g_1(\land\emptyset)=g_1(\land\\{0\\})=g_1(0)=0$$

but

$$\land g_1(\emptyset)=\land\emptyset=1$$

where for the first equation, \\(\emptyset\subseteq\\{0\\}\\) and for the second equation \\(\emptyset\subseteq\\{0,1\\}\\). So this is indeed consistent with Proposition 1.88.

I also want to understand what you meant by start as far up as we can when it comes to the empty set in #151. So going back to Alex's 2nd example in #147, it is true that \\(g_1\\) is not a right adjoint but it also does not preserve meet since:

$$g_1(\land\emptyset)=g_1(\land\\{0\\})=g_1(0)=0$$

but

$$\land g_1(\emptyset)=\land\emptyset=1$$

where for the first equation, \\(\emptyset\subseteq\\{0\\}\\) and for the second equation \\(\emptyset\subseteq\\{0,1\\}\\). So this is indeed consistent with Proposition 1.88.