On the equational de$nition of the least pre$xed point

We propose a method to axiomatize by equations the least pre$xed point of an order preserving function. We discuss its domain of application and show that the Boolean modal -calculus has a complete equational axiomatization. The method relies on the existence of a “closed structure” and its relationship to the equational axiomatization of Action Logic is made explicit. The implication operation of a closed structure is not monotonic in one of its variables; we show that the existence of such a term that does not preserve the order is an essential condition for de$ning by equations the least pre$xed point. We stress the interplay between closed structures and $xed point operators by showing that the theory of Boolean modal -algebras is not a conservative extension of the theory of modal -algebras. The latter is shown to lack the $nite model property. c © 2002 Elsevier Science B.V. All rights reserved.