Abstract
A protogroup is an ordered monoid in which each element a has both a left proto-inverse a ℓ such that a ℓa ≤ 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 ℓ and 1 ≤ a ra, rendering a ℓ 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 ℓℓ ≠ a in general. Free pregroups may be used to recognize the same sentences as free protogroups.
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Lambek, J. (1999). Type Grammar Revisited. In: Lecomte, A., Lamarche, F., Perrier, G. (eds) Logical Aspects of Computational Linguistics. LACL 1997. Lecture Notes in Computer Science(), vol 1582. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48975-4_1
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