We generalize the famous Tarski result by showing that: if X is a complete lattice, and f : X → X is an order-preserving mapping, then for all points x ∈ X, the limit superior and the limit inferior of the (possibly transfinite) sequence of iterations x, f(x), f2(x).., fβ(x),.. are fixed points of f. These limits are the sharp fixed-point bounds between which sufficiently large transfinite iterations are located.
- Tarski fixed-point theorem
ASJC Scopus subject areas
- Algebra and Number Theory
- Geometry and Topology
- Computational Theory and Mathematics