Tarski's fixpoint theorem states, that the fixpoints of a monotone operator on a complete lattice is a complete lattice. By consequence, we have a unique greatest fixpoint and unique least fixpoint for a monotone operator on a complete lattice.
The fixpoints can be unique but in general can be many.
My question would be, under which conditions can a monotone function have a unique fixpoint on a complete lattice? Is there some practical sufficient conditions to guarantee a unique fixpoint? It would useful to know this, because sometimes you have a monotone operator that specifies a property. It can be non-trivial to spell out whether it is the greatest fix point or the least fix point you really want to specify. In some cases, the two coincide, and you know to iterate from top or from bottom yields the same result and you would be happy to pick up the one that is more simple or more efficient.