Antonenko and Russyev independently have shown that any Mealy automaton with no cycles with exit—that is, where every cycle in the underlying directed graph is a sink component—generates a finite (semi)group, regardless of the choice of the production functions. Antonenko has proved that this constitutes a characterization in the non-invertible case and asked for the invertible case, which is p...

Let A be a commutative comodule algebra over a commutative bialgebra H . The group of invertible relative Hopf modules maps to the Picard group of A, and the kernel is described as a quotient group of the group of invertible grouplike elements of the coring A⊗H , or as a Harrison cohomology group. Our methods are based on elementary K-theory. The Hilbert 90 Theorem follows as a corollary. The p...

Due to the omission $\vphantom {\widehat X}}}}}}}$of a hypothesis from an elementary lemma in author’s paper \lq\lq Gleason parts and point derivations for uniform algebras with dense invertible group\rq\rq, some of proofs presented that are flawed. We prove here nevertheless, all results paper, exception one misstated lemma, correct. In process, we strengthen slightly paper.

A group-category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1-dimensional representations are the invertible simple objects. This paper gives a detailed exploration of “topological quantum field theories” for group-categories, in hopes of finding clues to a better u...

Several representations of the generalized Drazin inverse of a block matrix with a group invertible generalized Schur complement in Banach algebra are presented.