Let $X$ and $Y$ denote complete lattice, and suppose $f : X \rightarrow Y$ is a surjective order-homomorphism. Does $f$ necessarily preserve arbitrary suprema, therefore being a complete lattice homomorphism?
This seems too good to be true, but I've been unable to find a counterexample.
Let $X$ be two copies of the unit interval joined at their endpoints, with the order being the standard order on each. This gives rise to a nice poset, and I believe it is a complete lattice. The supremum of any set that meets both intervals is the unique maximal point, and the supremum of a set inside one of the intervals is the usual supremum inside the interval. Let $Y$ be the unit interval and let $f\colon X\to Y$ be the obvious map identifying both intervals. Then if you take the two points set consisting of the midpoints of each interval, the supremum does not map to the supremum. The supremum in the domain is the maximal point, and the supremum in the range is $1/2$.