J. Lambek. Bilinear logic in algebra and linguistics. In J.Y. Girard, Y. Lafont, and L. Regnier, editors, Advances in Linear Logic, volume 222 of London Math. Soc. Lec. Notes, page 43. Cambridge University Press, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... context of regular algebra by John Conway, who named them factors [8] More generally they are known as residuation operators [3, 13] and they feature in many di erent areas, such as software speci cation [19, 20] non commutative linear logic, and Lambek grammars in computational linguistics [24, 25]. The authors rst learned of these operators and their theory from Roland Backhouse [1, 14] For the present purposes, there are certain important facts about chip and chop. First, a language S is regular if and only if its number of chops (that is the cardinality of f S=R j R is any language g) ....

J. Lambek. Bilinear logic in algebra and linguistics. In J. Y. Girard, Y. Lafont, and L. Regnier, editors, Advances in Linear Logic, volume 222 of London Mathematical Society Lecture Note Series, pages 43-59. Cambridge University Press, 1995.

....then one should consider categories with two internal HOM s. Thus we should have adjunctions of the form: HOM(A Omega B; C) HOM(B;A Gammaffi C) HOM(A Omega B; C) HOM(A;C ffi Gamma B) This is the definition of biautonomous category, and is the basis of, for example, the Lambek calculus [24]. Analogously, to define a nonsymmetric analogue of categories with dualizing objects one needs two duals, A and A. The dualizing object for each will be the same. These will be subject to the isomorphisms: A ) A) A More specifically, a biautonomous category ....

J. Lambek, Bilinear Logic in Algebra and Linguistics, (1993).

....not only the autonomous structure (multiplicatives) but also additives and exponentials. ut Remark 3. 10 We can drop all symmetric from the results above and talk about monoidal (bi)closed categories, which are models of non commutative linear logic and also the syntactic calculus of Lambek [30]. The example below shows that a non commutative version of the phase semantics can be given as an instance of the subglueing construction. We also note that Shirasu [40] studied the glueing of monoidal (bi)closed lattices (called FL algebras ) along a monoidal meet semilattice morphism ( fringe ....

Lambek, J. (1995), Bilinear logic in algebra and linguistics, in "Advances in Linear Logic", pp. 43--59, Cambridge University Press.

....symmetric monoidal (closed) categories and adjunctions [6, 8, 5] and our approach based on such categorical models is likely to apply to many other linear type theories as well. In fact it is routine to modify our technique for non commutative linear logic and monoidal (bi)closed categories (see [17]) Furthermore, by combining our approach with Hyland and Tan s double glueing construction [23] see Example 5) we can deal with a classical linear type theory (MLL) These results, proofs and further category theoretic analysis are reported in the full paper [13] Also it might be fruitful to ....

Lambek, J. (1995), Bilinear logic in algebra and linguistics, in \Advances in Linear Logic", pp. 43-59, Cambridge University Press.

....B ) B A (exchange) fails, which suggests yet a more drastic modification, i.e. non commutative linear logic. By the way there is an interesting prefiguration of linear logic in the literature, namely Lambek s syntactic calculus, introduced in 1958 to cope with certain questions of linguistic, see [23], this volume. This system is based on a non commutative Omega , which in turn induces two linear implications. There would be no problems to enrich the system with additives and Phi, but the expressive power remains extremely limited. The missing items are exponentials and negation : I ....

J. Lambek. Bilinear logic in algebra and linguistics. In this volume, 1995.

....category is autonomous. In [10] the first author examines this category as a model of linear logic, and considers the representations of groups and Hopf algebras in such spaces. This theory leads to a large class of new models of commutative linear logic [19] noncommutative linear logic [4, 28], and braided linear logic [11] Proofs of all of the following results can be found in [6] and [10] Definition 5.1 Let V be a vector space over a discrete field k. A topology, on V is linear if it satisfies the following three properties: ffl Addition and scalar multiplication are ....

J. Lambek, Bilinear Logic in Algebra and Linguistics, preprint, (1993).

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J. Lambek. Bilinear logic in algebra and linguistics. In J.Y. Girard, Y. Lafont, and L. Regnier, editors, Advances in Linear Logic, volume 222 of London Math. Soc. Lec. Notes, page 43. Cambridge University Press, 1994.

No context found.

Lambek, J. (1995), `Bilinear logic in algebra and linguistics', in Girard et al. (eds.), Advances in Linear Logic, Cambridge, Cambridge University Press, 43-60.

No context found.

Lambek, J. (1995), 'Bilinear logic in algebra and linguistics', in Girard et al. (eds.), Advances in Linear Logic, Cambridge, Cambridge University Press.

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