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Completeness of the Lambek syntactic calculus
Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 1, pp. 193-219
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In this paper we prove that the Lambek syntactic calculus is complete with respect to free semigroup models, i. e. it derives exactly all identities concerning multiplication, left division, and right division of formal languages. 
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M. R. Pentus. Completeness of the Lambek syntactic calculus. Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 1, pp. 193-219. http://geodesic.mathdoc.fr/item/FPM_1999_5_1_a11/