Because of the isomorphism (X × A) → X ∼= X → (A → X), the transition structure of a deterministic automaton with state set X and with inputs from an alphabet A can be viewed both as an algebra and as a coalgebra. This algebra-coalgebra duality goes back to Arbib and Manes, who formulated it as a duality between reachability and observability, and is ultimately based on Kalman’s duality in syst...

Because of the isomorphism (X × A) → X ∼= X → (A → X), the transition structure of a deterministic automaton with state set X and with inputs from an alphabet A can be viewed both as an algebra and as a coalgebra. This algebra-coalgebra duality goes back to Arbib and Manes, who formulated it as a duality between reachability and observability, and is ultimately based on Kalman’s duality in syst...

Dana Scott’s model of λ-calculus was based on a limit construction which started from an algebra of a suitable endofunctor F and continued by iterating F . We demonstrate that this is a special case of the concept we call coalgebra relatively terminal w.r.t. the given algebra A. This means a coalgebra together with a universal coalgebra-toalgebra morphism into A. We prove that by iterating F co...

This paper generalizes the notion of local coefficients and fundamental groups of spaces to simplicial coalgebras. We define a Hopf algebra π1(C) from a simplicial coalgebra C as a generalization of a fundamental group, and show that a module overπ1(C) corresponds to a local coefficient of C. As a consequence, the Hoschild cohomology of a Hopf algebraH with a coefficient M coincides with the co...

Given a basic K-coalgebra C, we study the left Gabriel-valued quiver (CQ,Cd) of C by means of irreduciblemorphisms between indecomposable injective leftC-comodules and by means of the powers rad of the radical rad of the category C-inj of the socle-finite injective left C-comodules. Connections between the valued quiver (CQ,C d) of C and the valued quiver (CQ,Cd) of a colocalization coalgebra q...

The Chomsky hierarchy plays a prominent role in the foundations of theoretical computer science relating classes of formal languages of primary importance as well as the grammars and machine models that represent those language classes. It is well-known that regular languages can be captured by coalgebras. They form precisely the image of all finite coalgebras for the set functor X 7→ 2×X under...

Let $H$ be a bialgebra. $\sigma: H\otimes H\to A$ linear map, where $A$ is left $H$-comodule coalgebra, and an algebra with $H$-weak action $\triangleright$. $\tau: B$ $B$ right $\triangleleft$. In this paper, we improve the necessary conditions for two-sided crossed product $A\#^{\sigma} H~{^{\tau}\#} smash coproduct coalgebra $A\times H\times to form bialgebra (called double biproduct) such t...

We introduce the notion of idempotent radical class of module coalgebras over a bialgebra B. We prove that if R is an idempotent radical class of B-module coalgebras, then every B-module coalgebra contains a unique maximal B-submodule coalgebra in R. Moreover, a B-module coalgebra C is a member of R if, and only if, DB is in R for every simple subcoalgebra D of C. The collection of B-cocleft co...