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Entropic Hopf algebras and models of non-commutative logic

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Richard F. Blute, Francois Lamarche, Paul Ruet

We give a definition of categorical model for the multiplicative fragment
of non-commutative logic. We call such structures *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. Our first models are constructed
via the notion of a * 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 modeling a number of variants of multiplicative
linear logic, such as commutative, braided and cyclic. We extend these
ideas to the entropic setting by developing a new type of Hopf algebra,
which we call *entropic Hopf algebras*. We show that the category of
modules over an entropic Hopf algebra is an entropic category (possibly
after application of the Chu construction). Several examples are
discussed, based first on the notion of a *bigroup*. Finally the
Tannaka-Krein reconstruction theorem is extended to the entropic setting.

Keywords: Linear logic, monoidal categories, Hopf algebras.

2000 MSC: 03F07, 03F52, 18A15, 18D10, 57T05.

*Theory and Applications of Categories*, Vol. 10, 2002, No. 17, pp 424-460.

http://www.tac.mta.ca/tac/volumes/10/17/10-17.dvi

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