Complete Iterativity for Algebras with Effects

  • Stefan Milius
  • Thorsten Palm
  • Daniel Schwencke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5728)


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.


iterative algebra monad distributive law initial algebra terminal coalgebra 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Stefan Milius
    • 1
  • Thorsten Palm
    • 1
  • Daniel Schwencke
    • 1
  1. 1.Institut für Theoretische InformatikTechnische Universität BraunschweigBraunschweigGermany

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