Every variable-free logic program induces a PfPf -coalgebra on the set of atomic formulae in the program. The coalgebra p sends an atomic formula A to the set of the sets of atomic formulae in the antecedent of each clause for which A is the head. In an earlier paper, we identified a variable-free logic program with a PfPf -coalgebra on Set and showed that, if C(PfPf ) is the cofree comonad on ...

A lot of combinatorial objects have algebra and coalgebra structures posets are important objects. In this paper, we construct on the vector space spanned by posets. Firstly, associativity unitary property, prove that with conjunction product is a graded algebra. Then definition free algebra, free. Finally, coassociativity counitary unshuffle coproduct coalgebra.

We study serial coalgebras by means of their valued Gabriel quivers. In particular, Hom-computable and representation-directed coalgebras are characterized. The Auslander-Reiten quiver of a serial coalgebra is described. Finally, a version of Eisenbud-Griffith theorem is proved, namely, every subcoalgebra of a prime, hereditary and strictly quasi-finite coalgebra is serial. Introduction A syste...

We introduce the notion of a crossed product of an algebra by a coalgebra C, which generalises the notion of a crossed product by a bialgebra well-studied in the theory of Hopf algebras. The result of such a crossed product is an algebra which is also a right C-comodule. We find the necessary and sufficient conditions for two coalgebra crossed products be equivalent. We show that the two-dimens...

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 coalgebra for the power-set functor is prov...

A survey of parts of General Coalgebra is presented with applications to the theory of systems. Stress is laid on terminal coalgebras and coinduction as well as iterative algebras and iterative theories. 1. Preface: Is this Really an Introduction to Coalgebra? For my series of lectures on coalgebra on the preconference to CTCS 2002 I prepared a short text called “Introduction to Coalgebra” that...

In this work we study the induction (induced and coinduced)theory for Hopf group coalgebra. We define a substructure B of a Hopf group coalgebra H, called subHopf group coalgebra. Also, we have introduced the definition of Hopf group suboalgebra and group coisotropic quantum subgroup of H. The properties of the algebraic structure of the induced and coinduced are given. Moreover, a framework of...