Does lub-preservation imply monotonicity?

A function $latex f$ is continuous exactly when it is monotonic, i.e. $latex d\sqsubseteq d' \Rightarrow f(d)\sqsubseteq f(d')$, and lub-preserving, i.e. for any chain $latex d_0\sqsubseteq d_1 \sqsubseteq \dots$ we have $latex f(\bigsqcup_n d_n) = \bigsqcup_n f(d_n)$ (call this (1)) We can demonstrate that the first property is implied by the second (and is thus… Continue reading Does lub-preservation imply monotonicity?