Abstract
We prove that every symmetric residuated groupoid is embeddable in a boolean double residuated groupoid. Analogous results are obtained for other classes of algebras, e.g. (commutative) symmetric residuated semigroups, symmetric residuated unital groupoids, cyclic bilinear algebras. We also show that powerset algebras constructed in the paper preserve some Grishin axioms.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abrusci, V.M.: Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic. The Journal of Symbolic Logic 56(4), 1403–1451 (1991)
Białynicki-Birula, A., Rasiowa, H.: On the Representation of Quasi-Boolean Algebras. Bulletin de l’Académie Polonaise des Sciences 5, 259–261 (1957)
Bimbó, K., Dunn, J.M.: Generalized Galois Logics: Relational Semantics of Nonclassical Logical Calculi. CSLI Lecture Notes 188, Stanford (2008)
Buszkowski, W.: Categorial Grammars with Negative Information. In: Wansing, H. (ed.) Negation. A Notion in Focus, pp. 107–126. W. de Gruyter, Berlin (1996)
Buszkowski, W.: Interpolation and FEP for Logics of Residuated Algebras. Logic Journal of the IGPL 19(3), 437–454 (2011)
Dunn, J.M.: Generalized Ortho Negation. In: Wansing, H. (ed.) Negation. A Notion in Focus, pp. 3–26. W. de Gruyter, Berlin (1996)
Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics, vol. 151. Elsevier, Amsterdam (2007)
Grishin, V.N.: On a generalization of the Ajdukiewicz-Lambek system. In: Studies in Non-Commutative Logics and Formal Systems, Russian, Nauka, Moscow, pp. 315–343 (1983)
Kołowska-Gawiejnowicz, M.: Powerset Residuated Algebras and Generalized Lambek Calculus. Mathematical Logic Quarterly 43, 60–72 (1997)
Kurtonina, N., Moortgat, M.: Relational semantics for the Lambek-Grishin calculus. In: Ebert, C., Jäger, G., Michaelis, J. (eds.) MOL 10. LNCS, vol. 6149, pp. 210–222. Springer, Heidelberg (2010)
Lambek, J.: On the calculus of syntactic types. In: Jacobson, R. (ed.) Structure of Language and Its Mathematical Aspects, pp. 166–178. AMS, Providence (1961)
Lambek, J.: From Categorial Grammar to Bilinear Logic. In: Došen, K., Schröder-Heister, P. (eds.) Substructural Logics, pp. 207–237. Oxford University Press (1993)
Lambek, J.: Some lattice models of bilinear logic. Algebra Univeralis 34, 541–550 (1995)
Lambek, J.: Bilinear logic in algebra and linguistics. In: Girard, J.-Y., Lafont, Y., Regnier, L. (eds.) Advances in Linear Logic, pp. 43–59. Cambridge University Press, Cambridge (1995)
Lambek, J.: Bilinear logic and Grishin algebras. In: Orłowska, E. (ed.) Logic at Work, pp. 604–612. Physica-Verlag, Heidelberg (1999)
Moortgat, M.: Symmetries in natural language syntax and semantics: Lambek-Grishin calculus. In: Leivant, D., de Queiroz, R. (eds.) WoLLIC 2007. LNCS, vol. 4576, pp. 264–284. Springer, Heidelberg (2007)
Yetter, D.N.: Quantales and (non-commutative) linear logic. The Journal of Symbolic Logic 55(1), 41–64 (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Kołowska-Gawiejnowicz, M. (2014). On Canonical Embeddings of Residuated Groupoids. In: Casadio, C., Coecke, B., Moortgat, M., Scott, P. (eds) Categories and Types in Logic, Language, and Physics. Lecture Notes in Computer Science, vol 8222. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54789-8_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-54789-8_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-54788-1
Online ISBN: 978-3-642-54789-8
eBook Packages: Computer ScienceComputer Science (R0)