Knaster–Tarski theorem

The fixed point theorem of Tarski and Knaster, named after Bronisław Knaster and Alfred Tarski, is a mathematical theorem in the field of lattice theory.

Statement

Be a complete lattice, monotone, and the set of fixed points of in. Under these conditions and is also a complete lattice.

Idea of ​​proof

Is the supremum of operation, and the operation of infimum.

The following steps show that for any subset of an infimum and a supremum in supplies.

Consequences

One consequence often used is that of the existence of the smallest and largest fixed points of monotone functions with respect.