Abstract
The aim of this article is to present to universal algebraists a generalization of Boolean algebras which do not obey Gentzen's three structural rules. These so-calledGrishin algebras are models ofclassical bilinear propositional logic, a non-commutative version of linear logic which allows two negations. Examples in which these negations coincide are easy to come by, but examples in which they are distinct are more elusive. To this purpose, it was found necessary to generalize the notion of a group to that of alattice ordered monoid with adjoints. While the left inverse and the right inverse of a group element necessarily coincide this is not so for the left adjoint and the right adjoint of an element in a lattice ordered monoid.
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Dedicated to the memory of Alan Day
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Lambek, J. Some lattice models of bilinear logic. Algebra Universalis 34, 541–550 (1995). https://doi.org/10.1007/BF01181877
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DOI: https://doi.org/10.1007/BF01181877