Knaster-Tarski theorem

In mathematics,Knaster-Tarski theorem, named after Bronislaw KnasterAlfred Tarski, statesfollowing:

Let L becomplete latticelet f : L -> L be an order-preserving function. Thensetfixed pointsfLalsocomplete lattice.

Since complete lattices cannot be empty,theoremparticular guaranteesexistenceat least one fixed pointf,evenexistence ofleast (or greatest) fixed point. In many practical cases, this ismost important implication oftheorem.

For example,mathematical logic least fixed pointsfunctions on setsformulasusedcomputesemantics oflogic program. Sometimesmore specialized version oftheoremused, where Lassumedbelatticeall subsets ofcertain set ordered by subset inclusion. This reflectsfact thatmany applications only such latticesconsidered. One then usuallylooking forsmallest set that haspropertybeingfixed point offunction f.

References

• Alfred Tarski: A lattice-theoretical fixpoint theoremits applications. Pacific JournalMathematics, vol. 5 (1955), pp 285-309.