• Title, Summary, Keyword: Jonsson-Tarski algebra

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ON HOPF ALGEBRAS IN ENTROPIC JÓNSSON-TARSKI VARIETIES

  • ROMANOWSKA, ANNA B.;SMITH, JONATHAN D.H.
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.5
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    • pp.1587-1606
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    • 2015
  • 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 $J{\acute{o}}nsson$-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 $J{\acute{o}}nsson$-Tarski monoid of the generating algebra is cancellative. The problem of determining when the $J{\acute{o}}nsson$-Tarski monoid forms a group is open.