Iteration algebras are not finitely axiomatizable

Abstract

Algebras whose underlying set is a complete partial order and whose term-operations are continuous may be equipped with a least fixed point operation μx.t. The set of all equations involving the μ-operation which hold in all continuous algebras determines the variety of iteration algebras. A simple argument is given here reducing the ax-iomatization of iteration algebras to that of Wilke algebras. It is shown that Wilke algebras do not have a finite axiomatization. This fact implies that iteration algebras do not have a finite axiomatization, even by "hyperidentities". © Springer-Verlag Berlin Heidelberg 2000.