I have a poset P and a monotone function f from P to P. This function has the following special property:
if x ≥ f(y), there exists a y' ≥ y such that x = f(y')
The question is: is there a well-known name for this property, or a property that is closely related to this? It feels like some sort of 'connectivity' or 'continuity' or 'convexity' of the function to me. If one could hack up a topology out of the poset so that this is actually one of those concepts, that would be great.
