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The axioms of iteration theories, or iteration categories, capture theequational properties of fixed point operations in several computationallysignificant categories. Iteration categories may be axiomatized by the Conwayidentities and identities associated with finite automata. We show that inconjunction with the Conway identities, each identity associated with a finiteautomaton implies the identity associated with any input extension of theautomaton. We conclude that the Conway identities and the identities associatedwith the members of a subclass $\cQ$ of finite automata is complete foriteration categories iff for every finite simple group $G$ there is anautomaton $\bQ \in \cQ$ such that $G$ is a quotient of a group in the monoid$M(\bQ)$ of the automaton $\bQ$. We also prove a stronger result that concernsidentities associated with finite automata with a distinguished initial state.

Equational axioms associated with finite automata for fixed point operations in cartesian categories