This is a brief survey of some recent developments in the study of infinite dimensional Hopf algebras which are either noetherian or have finite Gelfand-Kirillov dimension. A number of open questions are listed.

We investigate the representation theory of finite sets. The correspondence functors are the functors from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. They have various specific properties which do not hold for other types of functors. In particular, if k is a field and if F is a correspondence functor, then F is finitely generate...

Abstract Taking residual finiteness as a starting point, we consider three related properties: weak subsemigroup separability, strong separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result this paper states that for finitely generated commutative semigroup S , conditions coincide are equivalent to every $\ma...