Some Remarks on Conway and Iteration Theories

We present an axiomatization of Conway theories which yields, as a corollary, a very concise axiomatization of iteration theories satisfying the functorial implication for base morphisms. It has been shown that most fixed point operations in computer science share the same equational properties. These equational properties are captured by the notion of iteration theories [1, 3]. Several axiomatizations of iteration theories have been presented in [2, 3]. For a recent overview, we refer to [6]. The axioms of iteration theories can be conveniently divided into two groups: axioms for Conway theories and the commutative identities. The commutative identities have later been simplified to the group identities or certain generalized power identities, cf. [4, 5, 7]. In this note, we provide further axiomatizations of Conway and iteration theories, see Corollaries 3,4 and 5. Iteration theories having a constructable fixed point operation (such as the theories of monotonic or continuous functions over a cpo or a complete lattice equipped with the least fixed point operation, or the theories of continuous functors over ω-categories equipped with the initial fixed point operation) usually satisfy the ‘functorial implication’ for base morphisms, see [2]. In fact, the commutative identities were introduced in [3] as a strictly weaker, but fully equational substitute for the functorial implication for (surjective) base morphisms, which are however still sufficient for completeness. As a corollary of our results, we obtain a simple axiomatization of iteration theories with a functorial dagger for base morphisms, cf. Corollary 6. We assume familiarity with Conway and iteration theories and closely follow the terminology and notation in [2]. Proposition 1 Let T be a preiteration theory. Then T is a Conway theory iff T satisfies the (base) parameter, fixed point, permutation and double dagger identities as well as the following identity: (1n ⊕ 0m) · 〈f · (1n ⊕ 0m ⊕ 1p), g〉 † = f , (1) ∗Partially supported by grant no. ANN 110883 from the National Foundation of Hungary for Scientific Research.