Anna B. Romanowska and Jonathan D. H. Smith Warsaw University of Technology, Iowa State University

Abstract : Comonoid, bi-algebra, and Hopf algebra structures are studied within the universal-algebraic context of entropic varieties. Attention focuses on the behavior of setlike and primitive elements. It is shown that entropic J\'{o}nsson-Tarski varieties provide a natural universal-algebraic setting for primitive elements and group quantum couples (generalizations of the group quantum double). Here, the set of primitive elements of a Hopf algebra forms a Lie algebra, and the tensor algebra on any algebra is a bi-algebra. If the tensor algebra is a Hopf algebra, then the underlying J\'{o}nsson-Tarski monoid of the generating algebra is cancellative. The problem of determining when the J\'{o}nsson-Tarski monoid forms a group is open.