A pregroup is a partially ordered monoid endowed with two unary operations called left and right adjunction. Pregroups were recently introduced to help with natural language processing, as we illustrate here by looking at small fragments of three modern European languages. As it turns out, the apparently new algebraic concept of a pregroup had been around for some time in recreational mathematics, although not under this name.

Grammar can be formulated as a kind of substructural propositional logic. In support of this claim, we survey bare Gentzen style deductive systems and two kinds
of non-commutative linear logic: intuitionistic and compact bilinear logic. We also glance at their categorical refinements