Corecursive Algebras, Corecursive Monads and Bloom Monads

An algebra is called corecursive if from every coalgebra a unique coalgebra-to-algebra homomorphism exists into it. We prove that free corecursive algebras are obtained as coproducts of the terminal coalgebra (considered as an algebra) and free algebras. The monad of free corecursive algebras is proved to be the free corecursive monad, where the concept of corecursive monad is a generalization of Elgot's iterative monads, analogous to corecursive algebras generalizing completely iterative algebras. We also characterize the Eilenberg-Moore algebras for the free corecursive monad and call them Bloom algebras

Presenting Distributive Laws

Distributive laws of a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specification of well-behaved structural operational semantics and, more recently, also for enhancements of the bisimulation proof method. If T is a free monad, then such distributive laws correspond to simple natural transformations. However, when T is not free it can be rather difficult to prove the defining axioms of a distributive law. In this paper we describe how to obtain a distributive law for a monad with an equational presentation from a distributive law for the underlying free monad. We apply this result to show the equivalence between two different representations of context-free languages

On Corecursive Algebras for Functors Preserving Coproducts

For an endofunctor H on a hyper-extensive category preserving countable coproducts we describe the free corecursive algebra on Y as the coproduct of the terminal coalgebra for H and the free H-algebra on Y. As a consequence, we derive that H is a cia functor, i.e., its corecursive algebras are precisely the cias (completely iterative algebras). Also all functors H(-) + Y are then cia functors. For finitary set functors we prove that, conversely, if H is a cia functor, then it has the form H = W times (-) + Y for some sets W and Y

The Sierpinski Carpet as a Final Coalgebra

We advance the program of connections between final coalgebras as sources of circularity in mathematics and fractal sets of real numbers. In particular, we are interested in the Sierpinski carpet, taking it as a fractal subset of the unit square. We construct a category of square sets and an endofunctor on it which corresponds to the operation of gluing copies of a square set along segments. We show that the initial algebra and final coalgebra exist for our functor, and that the final coalgebra is bi-Lipschitz equivalent to the Sierpinski carpet. Along the way, we make connections to topics such as the iterative construction of initial algebras as colimits, corecursive algebras, and the classic treatment of fractal sets due to Hutchinson.Comment: In Proceedings ACT 2021, arXiv:2211.0110