# Algebra and Coalgebra in Computer Science: Third by Alexander Kurz, Marina Lenisa

By Alexander Kurz, Marina Lenisa

This publication constitutes the complaints of the 3rd foreign convention on Algebra and Coalgebra in desktop technology, CALCO 2009, shaped in 2005 by way of becoming a member of CMCS and WADT. This yr the convention was once held in Udine, Italy, September 7-10, 2009. The 23 complete papers have been conscientiously reviewed and chosen from forty two submissions. they're awarded including 4 invited talks and workshop papers from the CALCO-tools Workshop. The convention used to be divided into the next periods: algebraic results and recursive equations, idea of coalgebra, coinduction, bisimulation, stone duality, online game idea, graph transformation, and software program improvement recommendations.

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In this paper we consider cias in the two categories Set and SetM . To distinguish ¯ we have the ordinary cias for an endofunctor H on Set from those for a lifting H following ¯ : SetM → SetM a Kleisli-cia. 7. 8. If we spell out the definition of a cia in SetM , we see that a flat equation morphism is a map e : X → M (HX + A), and a solution of e in the algebra α : HA → M A is a map e† : X → M A such that M(λ·He† +ηA ) e† = (X −→ M (HX + A) −−−−−−−−−→ M (M HA + M A) e μX+A ·M[Minl,Minr] μA ·[α,ηA ] −−−−−−−−−−−−−→ M (HA + A) −−−−−−→ M A) This means that the algebra (A, α) as well as the recursive equation e and its solution are “effectful”.

14. Every Kleisli-cia is a λ-cia. Proof. Let α : HA → M A be a Kleisli-cia and let e : X → HX + M A be an ¯ + A as M -equation morphism. We form a flat equation morphism e¯ : X ◦ / HX follows: ¯ +εA Je ¯ + M A idHX / HX ¯ + A) . 9) that a morphism s : X ◦ / A is a solution of e¯ in the Kleisli-cia A iff it is a solution of e. Since the former exists uniquely, so does the latter; thus (A, α) is a λ-cia. 15. 14 does not hold in general. Let H = Id, let M X = X ∗ be the monad of finite lists on X, and let λ = id : M ⇒ M .

1. Stateful computations with nontermination: T A = S → (1 + A × S), where S is a fixed set of states. 2. Nondeterminism: T A = P(A), where P is the covariant power set functor. 3. Exceptions: T A = A + E, where E is a fixed set of exceptions. 4. Interactive input: T A is the smallest fixed point of γ → A + (U → γ), where U is a set of input values. 5. Interactive output: T A is the smallest fixed point of γ → A + (U × γ), where U is a set of output values. 6. Nondeterministic stateful computation: T A = S → P(A × S).