Abstract
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 of M. 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 of H (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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Milius, S., Palm, T., Schwencke, D. (2009). Complete Iterativity for Algebras with Effects. In: Kurz, A., Lenisa, M., Tarlecki, A. (eds) Algebra and Coalgebra in Computer Science. CALCO 2009. Lecture Notes in Computer Science, vol 5728. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03741-2_4
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DOI: https://doi.org/10.1007/978-3-642-03741-2_4
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