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Inÿnite trees form a free completely iterative theory over any given signature—this fact, proved by Elgot, Bloom and Tindell, turns out to be a special case of a much more general categorical result exhibited in the present paper. We prove that whenever an endofunctor H of a category has ÿnal coalgebras for all functors H () + X , then those coalgebras, TX… (More)

For deterministic systems, expressed as coalgebras over polynomial functors, every tree t (an element of the final coalgebra) turns out to represent a new coalgebra A t. The universal property of these coalgebras, resembling freeness, is that for every state s of every system S there exists a unique coalgebra homomorphism from a unique A t which takes the… (More)

- Jirí Adámek
- 1991

A concept of equation morphism is introduced for every endofuctor F of a cocomplete category C. Equationally defined classes of F –algebras for which free algebras exist are called varieties. Every variety is proved to be monadic over C, and conversely, every monadic category is equivalent to a variety. And the Birkhoff Variety Theorem is proved for "… (More)

The category Class of classes and functions is proved to have a number of properties suitable for algebra and coalgebra: every endofunctor has an initial algebra and a terminal coalgebra, the categories of algebras and coalgebras are complete and cocomplete, and every endofunctor generates a free completely iterative monad. A description of a terminal… (More)

Denotational semantics can be based on algebras with additional structure (order, metric, etc.) which makes it possible to interpret recursive specifications. It was the idea of Elgot to base denotational semantics on iterative theories instead, i. e., theories in which abstract recursive specifications are required to have unique solutions. Later Bloom… (More)