Fixpoints in Complete Lattices 1 Piotr Rudnicki University

Theorem (5) states that if an iterate of a function has a unique fixpoint then it is also the fixpoint of the function. It has been included here in response to P. Andrews claim that such a proof in set theory takes thousands of lines when one starts with the axioms. While probably true, such a claim is misleading about the usefulness of proof-checking systems based on set theory. Next, we prove the existence of the least and the greatest fixpoints for ⊆-monotone functions from a powerset to a powerset of a set. Scheme Knaster is the Knaster theorem about the existence of fixpoints, cf. [14]. Theorem (11) is the Banach decomposition theorem which is then used to prove the Schröder-Bernstein theorem (12) (we followed Paulson’s development of these theorems in Isabelle [16]). It is interesting to note that the last theorem when stated in Mizar in terms of cardinals becomes trivial to prove as in the Mizar development of cardinals the ≤ relation is synonymous with ⊆. Section 3 introduces the notion of the lattice of a lattice subset provided the subset has lubs and glbs. The main theorem of Section 4 is the Tarski theorem (43) that every monotone function f over a complete lattice L has a complete lattice of fixpoints. As the consequence of this theorem we get the existence of the least fixpoint equal to f(⊥L) for some ordinal β with cardinality not bigger than the cardinality of the carrier of L, cf. [14], and analogously the existence of the greatest fixpoint equal to f(⊤L). Section 5 connects the fixpoint properties of monotone functions over complete lattices with the fixpoints of ⊆-monotone functions over the lattice of subsets of a set (Boolean lattice).