We investigate functors between abelian categories having usomor-phic left and right adjoint functors (these functors are called Frobenius Functors). They are characterized for categories of modules and categories of comodules. We give some applications in coalgebras and Hopf modules. In particular, we introduce the notion of Frobenius homomorphism of coalgebras. The set of isomorphism classes of Frobenius functors between quite general Grothendieck categories is endowed with an abelian group structure. This gives a functorial notion of Grothendieck group which behaves satisfactorily.

In this paper, we propose a new way of providing cyclic structures to convolutional codes. We define the skew cyclic convolutional codes as left ideals of a quotient ring of a suitable non-commutative polynomial ring. In contrast to the previous approaches to cyclicity for convolutional codes, we use Ore polynomials with coefficients in a field (the rational function field over a finite field), so their arithmetic is very well known and we may proceed similarly to cyclic block codes. In particular, we show how to obtain easily skew cyclic convolutional codes of a given dimension, and we compute an idempotent generator of the code and its dual.

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 nice enough 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.