**JL1:** Let \\(U\\) be an upset, and let \\(x \in \operatorname{\uparrow}(U)\\). Then there is some \\(y \in U\\) such that \\(y \le x\\). Since \\(U\\) is an upset, this means that \\(x \in U\\). Therefore, \\(U = \operatorname{\uparrow}(U)\\). So every upset \\(U\\) is of the form \\(\operatorname{\uparrow}(S)\\) for some \\(S\\); at the very least, take \\(S\\) to be \\(U\\) itself, forgetting its order structure.

I think there’s an adjoint relationship happening here too, between \\(\operatorname{\uparrow}\\) and the forgetful function from posets to sets. In fact, \\(\operatorname{\uparrow}\\) also seems to be a closure operator, which as someone else noted is what monads are for posets.