Group Axioms for Iteration

Abstract

Iteration theories provide a sound and complete axiomatization of the equational properties of the iteration (or fixed point) operation in many models of theoretical computer science including ordered and metric structures, trees and synchronization trees. All known equational axiomatizations of iteration theories consist of a small set of equational axioms for Conway theories and a complicated equation scheme, the commutative identity. Here we associate an identity with each finite semigroup. We prove that the set consisting of the Conway identities and the group identities associated with the finite (simple) groups is complete. Moreover, we prove that the Conway identities and a subcollection of the semigroup identities associated with a subclass of the finite semigroups is complete iff each finite (simple) group divides one of the semigroups in the subclass. We also formulate a conjecture and study its consequences. The results are a generalization of Krob's axiomatization of the equational theory of the regular sets.