Entropic Hopf Algebras and Models of Non Commutative Logic

Richard F. Blute 1 François Lamarche 2 Paul Ruet
2 CALLIGRAMME - Linear logic, proof networks and categorial grammars
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : We give a definition of categorical models for the multiplicative fragment of non-commutative logic, which we call {\it entropic categories}. We demonstrate the soundness and completeness of our axiomatization with respect to cut-elimination. We then focus on several methods of building entropic categories. The first method is constructed via the notion of a {\it partial bimonoid} acting on a cocomplete category. We also explore an entropic version of the Chu construction, and apply it in this setting. It has recently been demonstrated that Hopf algebras provide an excellent framework for modelling a number of variants of multiplicative linear logic, such as commutative, braided and cyclic. We extend these ideas to the entropic setting by developping a new type of Hopf algebra, which we call {\it entropic Hopf algebras}. We show that the category of modules over an entropic Hopf algebra is an entropic category.Several examples are discussed, based first on the notion of a {\it bigroup}.
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Submitted on : Tuesday, September 26, 2006 - 2:49:54 PM
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Richard F. Blute, François Lamarche, Paul Ruet. Entropic Hopf Algebras and Models of Non Commutative Logic. Theory and Applications of Categories, Mount Allison University, 2002, 10 (17), pp.424-460. ⟨inria-00100711⟩



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