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**1 - 5**of**5**### Contents lists available at ScienceDirect

"... Information and Computation journal homepage:www.elsevier.com/locate/ic Complete sets of cooperations ..."

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Information and Computation journal homepage:www.elsevier.com/locate/ic Complete sets of cooperations

### London, United Kingdom

"... The structure map turning a set into the carrier of a final coalgebra is not unique. This fact is well-known but commonly elided. In this paper we argue that any such concrete representation of a set as a final coalgebra is potentially interesting on its own. We discuss several examples, in particul ..."

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The structure map turning a set into the carrier of a final coalgebra is not unique. This fact is well-known but commonly elided. In this paper we argue that any such concrete representation of a set as a final coalgebra is potentially interesting on its own. We discuss several examples, in particular, we consider different coalgebra structures that turn the set of infinite streams into the carrier of a final coalgebra. After that we focus on coalgebra structures that are made up using so-called cooperations. We say that a collection of cooperations is complete for a given set X if it gives rise to a coalgebra structure that turns X into the carrier set of a subcoalgebra of a final coalgebra. Any complete set of cooperations yields a coalgebraic proof and definition principle. We exploit this fact and devise a general definition scheme for constants and functions on a set X that is parametrically in the choice of the complete set of cooperations for X. Key words: Coalgebra, coinduction, infinite data structures, hidden algebra. 1

### www.elsevier.com/locate/entcs Observational Coalgebras and Complete Sets of Co-operations

"... In this paper we introduce the notion of an observational coalgebra structure and of a complete set of co-operations. We demonstrate in various example the usefulness of these notions, in particular, we show how they give rise to coalgebraic proof and definition principles. Keywords: Coalgebra, Coin ..."

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In this paper we introduce the notion of an observational coalgebra structure and of a complete set of co-operations. We demonstrate in various example the usefulness of these notions, in particular, we show how they give rise to coalgebraic proof and definition principles. Keywords: Coalgebra, Coinduction, infinite data structures, Hidden Algebra.

### An introduction to (co)algebra and (co)induction

"... Algebra is a well-established part of mathematics, dealing with sets with operations satisfying certain properties, like groups, rings, vector spaces, etcetera. Its results are essential throughout mathematics and other sciences. Universal algebra is a part of algebra in which algebraic structures a ..."

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Algebra is a well-established part of mathematics, dealing with sets with operations satisfying certain properties, like groups, rings, vector spaces, etcetera. Its results are essential throughout mathematics and other sciences. Universal algebra is a part of algebra in which algebraic structures are studied at a high