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**Additional info for Algebraic Informatics: 4th International Conference, CAI 2011, Linz, Austria, June 21-24, 2011. Proceedings**

**Sample text**

Let Σ be a constructive signature, I be the S-sorted set with Is = ∅ for all s ∈ S, ini be the unique S-sorted function from I to HΣ (I) and ∼s n n is the equivalence closure of {(a, HΣ (ini)(a)) | a ∈ HΣ (I)s , n ∈ N}. By Thm. 3, the following Σ-algebra A is initial: For all s ∈ S and f : e → s ∈ F , As = ( n HΣ (I)s )/∼s and f A = ini (HΣ ) ◦ ιf . n∈N Let B be a Σ-algebra, β0 be the unique S-sorted function from I to B and n+1 (I)s → Bs . for all n ∈ N and s ∈ S, βn+1,s = [f B ◦ Fe (βn,s )]f :e→s∈F : HΣ B The unique Σ-homomorphism fold : A → B is the unique S-sorted function satisfying fold B ◦ nat∼ = [βn ]n∈N .

Hence we add the identity on N and show that the desired property for fact and id simultaneously. The corresponding equations read as follows: fact, id (0) = (1, 0), fact , id (n + 1) = (fact (n) ∗ (id(n) + 1), id(n) + 1). We choose the product adjunction ((_, _) : Set → Set2 , × : Set2 → Set, λA. idA , idA , (π1 , π2 )). The latter equations imply that the kernel of the Set-extension (fact , id)# = fact , id : N → N2 of (fact , id) : (N, N) → (N, N) is compatible with 0 and succ. Hence (fact , id) is Nat-recursive and by Prop.

But not any member of such a class of models admits the application of powerful methods to operate on and reason about it. Among the members of an algebraic class it is the initial one, among those of a coalgebraic class it is the final one that the modeling should aim at. Initial algebras enable recursion and induction. Final coalgebras enable corecursion and coinduction. Twenty years ago algebraic modeling was mainly algebraic speciﬁcation and thus initial and free algebras were the main objects of interest [18,17,14,9], although hidden algebra and final semantics approaches [21,30,16,27,47,48] already tended to the coalgebraic view (mostly in terms of greatest quotients of initial models).