A non-unital generalization of weak bialgebra is proposed with a multiplier-valued comultiplication. Certain canonical subalgebras of the multiplier algebra (named the 'base algebras') are shown to carry coseparable co-Frobenius coalgebra structures. Appropriate modules over a weak multiplier bialgebra are shown to constitute a monoidal category via the (co)module tensor product over the base (co)algebra. The relation to Van Daele and Wang's (regular and arbitrary) weak multiplier Hopf algebra is discussed. Date

This series is aimed at publishing work dealing with the definition, development and application of fundamental theory and methodology, computational and algorithmic implementations and comprehensive empiric al studies in mathematical modelling. Work on new mathematics inspired by the construction of mathematical models, combining theory and experiment and furthering the understanding of the systems being modelled are particularly we1comed.

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We develop some basic functorial techniques for the study of the categories of comodules over corings. In particular, we prove that the induction functor stemming from every morphism of corings has a left adjoint, called ad-induction functor. This construction generalizes the known adjunctions for the categories of Doi-Hopf modules and entwined modules. The separability of the induction and ad-induction functors are characterized, extending earlier results for coalgebra and ring homomorphisms, as well as for entwining structures.2000 Mathematics Subject Classification: 16W30.1. Introduction. The notion of separable functor was introduced by Nȃstȃsescu et al. [12], where some applications for group-graded rings were done. This notion fits satisfactorily to the classical notion of separable algebra over a commutative ring. Every separable functor between abelian categories encodes a Maschke's theorem, which explains the interest concentrated in this notion within the module-theoretical developments in recent years. Thus, separable functors have been investigated in the framework of coalgebras [8], graded homomorphisms of rings [9,10], Doi-Koppinen modules [6,7], or finally, entwined modules [4,5]. These situations are generalizations of the original study of the separability for the induction and restriction of scalars functors associated to a ring homomorphism done in [12]. It turns out that all the aforementioned categories of modules are instances of comodule categories over suitable corings [3]. In fact, the separability of some fundamental functors relating the category of comodules over a coring and the underlying category of modules has been studied in [3]. Thus, we can expect that the characterizations obtained in [4] of the separability of the induction functor associated to an admissible morphism of entwining structures and its adjoint generalize to the corresponding functors stemming from a homomorphism of corings. This is done in this paper.To state and prove the separability theorems, a basic theory of functors between categories of comodules has been developed in this paper, making the arguments independent from the Sweedler's sigma-notation. The plan here is to use purely categorical methods which could be easily adapted to more general developments of the theory. These methods had been sketched in [1,2] in the framework of coalgebras over commutative rings and are expounded in Sections 2, 3, and 4. In Section 5, a notion of homomorphism of corings is given, which leads to a pair of adjoint functors (the induction functor and its adjoint, called here ad-induction functor). The morphisms of entwining structures [4] are instances of homomorphisms of corings in our setting. Finally, the separability of these functors is characterized.We use essentially the categorical terminology of [16], with the exception of the

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In this paper we extend the theory of serial and uniserial ÿnite dimensional algebras to coalgebras of arbitrary dimension. Nakayama-Skorniakov Theorems are proved in this new setting and the structure of such coalgebras is determined up to Morita-Takeuchi equivalences. Our main structure theorem asserts that over an algebraically closed ÿeld k the basic coalgebra of a serial indecomposable coalgebra is a subcoalgebra of a path coalgebra k where the quiver is either a cycle or a chain (ÿnite or inÿnite). In the uniserial case, is either a single point or a loop. For cocommutative coalgebras, an explicit description is given, serial coalgebras are uniserial and these are isomorphic to a direct sum of subcoalgebras of the divided power coalgebra.

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