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 <TEX>$J{\acute{o}}nsson$</TEX>-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 <TEX>$J{\acute{o}}nsson$</TEX>-Tarski monoid of the generating algebra is cancellative. The problem of determining when the <TEX>$J{\acute{o}}nsson$</TEX>-Tarski monoid forms a group is open.
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