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Abstract

A protogroup is an ordered monoid in which each element a has both a left proto-inverse a such that a a ≤ 1 and a right proto-inverse a r such that aa r ≤ 1. We explore the assignment of elements of a free protogroup to English words as an aid for checking which strings of words are well-formed sentences, though ultimately we may have to relax the requirement of freeness. By a pregroup we mean a protogroup which also satisfies 1 ≤ aa and 1 ≤ a r a, rendering a a left adjoint and a r a right adjoint of a. A pregroup is precisely a poset model of classical non-commutative linear logic in which the tensor product coincides with it dual. This last condition is crucial to our treatment of passives and Wh-questions, which exploits the fact that a ℓℓa in general. Free pregroups may be used to recognize the same sentences as free protogroups.

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