Complete Iterativity for Algebras with Effects
نویسندگان
چکیده
Completely iterative algebras (cias) are those algebras in which recursive equations have unique solutions. In this paper we study complete iterativity for algebras with computational effects (described by a monad). First, we prove that for every analytic endofunctor on Set there exists a canonical distributive law over any commutative monad M , hence a lifting of that endofunctor to the Kleisli category ofM . Then, for an arbitrary distributive law λ of an endofunctor H on Set over a monad M we introduce λ-cias. The cias for the corresponding lifting ofH (called Kleisli-cias) form a full subcategory of the category of λ-cias. For various monads of interest we prove that free Kleisli-cias coincide with free λ-cias, and these free algebras are given by free algebras for H . Finally, for three concrete examples of monads we prove that Kleisli-cias and λ-cias coincide and give a characterisation of those algebras.
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