Abstract
Lambek logics are substructural logics related to the Syntactic Calculus of Lambek [17]. In this paper we prove several representation theorems for algebras of Lambek logics (residuated semigroups, residuated monoids and others) with respect to certain algebras of binary relations. First results of this kind were obtained by Andréka and Mikulás [1], using a method of labeled graphs. Other results were proved in [9,28], using a method of labeled formulas. In the present paper, we prove these and other results, using a construction of chains of partial representations; this idea was announced earlier in the abstract [5]. We also provide a simpler construction which works for right and left pregroups [8].
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References
Andréka, H., Mikulás, S.: Lambek Calculus and Its Relational Semantics. Completeness and Incompleteness. Journal of Logic, Language and Information 3(1), 1–37 (1994)
van Benthem, J.: Language in Action. Categories, Lambdas and Dynamic Logic. North Holland, Amsterdam (1991)
Buszkowski, W.: Completeness Results for Lambek Syntactic Calculus. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 32, 13–28 (1986)
Buszkowski, W.: Algebraic Structures in Categorial Grammar. Theoretical Computer Science 199, 1–24 (1998)
Buszkowski, W.: More on embeddings of residuated semigroups into algebras of relations (abstract). In: Orłowska, E., Szałas, A. (eds.) Relational Methods in Logic, Algebra and Computer Science, Warsaw, pp. 33–36 (1998)
Buszkowski, W.: Finite Models of Some Substructural Logics. Mathematical Logic Quarterly 48, 63–72 (2002)
Buszkowski, W.: Lambek Grammars Based on Pregroups. In: de Groote, P., Morrill, G., Retoré, C. (eds.) LACL 2001. LNCS (LNAI), vol. 2099, pp. 95–109. Springer, Heidelberg (2001)
Buszkowski, W.: Pregroups: Models and Grammars. In: de Swart, H. (ed.) RelMiCS 2001. LNCS, vol. 2561, pp. 35–49. Springer, Heidelberg (2002)
Buszkowski, W., Kołowska-Gawiejnowicz, M.: Representation of Residuated Semigroups in Some Algebras of Relations (The Method of Canonical Models). Fundamenta Informaticae 31, 1–12 (1997)
Dunn, J.M.: Partial gaggles applied to logics with restricted structural rules. In: [13], pp. 63–108
Došen, K.: Sequent Systems and Groupoid Models. Studia Logica 47, 353–386 (1988), 48, 41–65 (1989)
Došen, K.: A Brief Survey of Frames for The Lambek Calculus. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 38, 179–187 (1992)
Došen, K., Schroeder-Heister, P. (eds.): Substructural Logics. Oxford University Press, Oxford (1993)
Girard, J.Y.: Linear logic. Theoretical Computer Science 50, 1–102 (1987)
Hoare, C.A.R., Jifeng, H.: The weakest prespecification. Fundamenta Informaticae 9, 51–84, 217–262 (1986)
Kurtonina, N.: Frames and Labels. A Modal Analysis of Categorial Inference. Ph.D. Thesis, University of Utrecht (1995)
Lambek, J.: The mathematics of sentence structure. American Mathematical Monthly 65, 154–170 (1958)
Lambek, J.: On the calculus of syntactic types. In: Jakobson, R. (ed.) Structure of Language and Its Mathematical Aspects, pp. 166–178. AMS, Providence (1961)
Lambek, J.: Type Grammars Revisited. In: Lecomte, A., Perrier, G., Lamarche, F. (eds.) LACL 1997. LNCS (LNAI), vol. 1582, pp. 1–27. Springer, Heidelberg (1999)
MacCaull, W., Orłowska, E.: Correspondence Results for Relational Proof Systems with Application to the Lambek Calculus. Studia Logica 71, 389–414 (2002)
Moortgat, M.: Categorial Type Logics. In: van Benthem, J., ter Meulen, A. (eds.) Handbook of Logic and Language, pp. 93–177. Elsevier, Amsterdam (1997)
Ono, H.: Semantics for Substructural Logics. In: [13], pp. 259–291
Orłowska, E.: Relational Interpretation of Modal Logics. In: Andréka, H., Monk, D., Nemeti, I. (eds.) Algebraic Logic, pp. 443–471. North Holland, Amsterdam (1988)
Orłowska, E.: Relational proof system for relevant logic. Journal of Symbolic Logic 57, 1425–1440 (1992)
Pentus, M.: Models for The Lambek Calculus. Annals of Pure and Applied Logic 75, 179–213 (1995)
Pratt, V.: Action Logic and Pure Induction. In: van Eijck, J. (ed.) JELIA 1990. LNCS (LNAI), vol. 478, pp. 97–120. Springer, Heidelberg (1991)
Restall, G.: An Introduction to Substructural Logics. Routledge, London (2000)
Szczerba, M.: Relational Models for The Nonassociative Lambek Calculus. In: Orłowska, E., Szałas, A. (eds.) Relational Methods for Computer Science Applications, pp. 149–159. Physica Verlag, Heidelberg (2001)
Venema, Y.: Tree models and (labelled) categorial grammars. Journal of Logic, Language and Information 5, 253–277 (1996)
Yetter, D.N.: Quantales and (non-commutative) linear logic. Journal of Symbolic Logic 55, 41–64 (1996)
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Buszkowski, W. (2003). Relational Models of Lambek Logics. In: de Swart, H., Orłowska, E., Schmidt, G., Roubens, M. (eds) Theory and Applications of Relational Structures as Knowledge Instruments. Lecture Notes in Computer Science, vol 2929. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24615-2_9
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