Let F be a monotone
operator on the complete lattice L into itself. Tarski’s lattice theoretical fixed point
theorem states that the set of fixed points of F is a nonempty complete lattice for
the ordering of L. We give a constructive proof of this theorem showing
that the set of fixed points of F is the image of L by a lower and an upper
preclosure operator. These preclosure operators are the composition of lower and
upper closure operators which are defined by means of limits of stationary
transfinite iteration sequences for F. In the same way we give a constructive
characterization of the set of common fixed points of a family of commuting
operators. Finally we examine some consequences of additional semi-continuity
hypotheses.
