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Cartesian isomorphisms are symmetric monoidal: A justification of linear logic

Published online by Cambridge University Press:  12 March 2014

Kosta Došen  and
Zoran Petrić
Kosta Došen
Affiliation:
Irit, University of Toulouse III, 31062 Toulouse Cedex, France Mathematical Institute, P. O. Box 367, 11001 Belgrade, Yugoslavia
Zoran Petrić
Affiliation:
University of Belgrade, Faculty of Mining and Geology, Djušina 7, 11000 Belgrade, Yugoslavia
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Abstract

It is proved that all the isomorphisms in the cartesian category freely generated by a set of objects (i.e., a graph without arrows) can be written in terms of arrows from the symmetric monoidal category freely generated by the same set of objects. This proof yields an algorithm for deciding whether an arrow in this free cartesian category is an isomorphism.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 1 , March 1999 , pp. 227 - 242
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1]Bruce, K. B., Cosmo, R. Di, and Longo, G., Provable isomorphisms of types, Mathematical Structures in Computer Science, vol. 2 (1992), pp. 231–247.CrossRefGoogle Scholar
[2]Cosmo, R. Di, Isomorphisms of types: From λ-calculus to information retrieval and language design, Birkhäuser, Boston, 1995.CrossRefGoogle Scholar
[3]Došen, K., Logical constants as punctuation marks, Notre Dame Journal of Formal Logic, vol. 30 (1989), pp. 362–381, slightly amended version in What is a logical system? (D. M. Gabbay, editor), Oxford University Press, Oxford, 1994, pp. 273–296.CrossRefGoogle Scholar
[4]Došen, K., Modal translations in substructural logics, Journal of Philosophical Logic, vol. 21 (1992), pp. 283–336.CrossRefGoogle Scholar
[5]Došen, K. and Petrić, Z., Modal functional completeness, Proof theory of modal logic (Wansing, H., editor), Kluwer, Dordrecht, 1996, pp. 167–211.CrossRefGoogle Scholar
[6]Došen, K. and Schroeder-Heister, P. (editors), Substructural logics, Oxford University Press, Oxford, 1993.Google Scholar
[7]Gentzen, G., Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie, Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften (N.S.), vol. 4 (1938), pp. 19–44, English translation in The collected papers of Gerhard Gentzen (M. E. Szabo, editor), North-Holland, Amsterdam, 1969, pp. 252–286.Google Scholar
[8]Kelly, G. M., Many-variable functorial calculus I, Coherence in categories (Lane, S. Mac, editor), Lecture Notes in Mathematics, no. 281, Springer-Verlag, Berlin, 1972, pp. 66–105.CrossRefGoogle Scholar
[9]Kelly, G. M., An abstract approach to coherence, Coherence in categories (Lane, S. Mac, editor), Lecture Notes in Mathematics, no. 281, Springer-Verlag, Berlin, 1972, pp. 106–147.CrossRefGoogle Scholar
[10]Mac Lane, S., Categories for the working mathematician, Springer-Verlag, Berlin, 1971.CrossRefGoogle Scholar
[11]Maehara, S., Eine Darstellung der intuitionistischen Logik in der klassischen, Nagoya Mathematical Journal, vol. 7 (1954), pp. 45–64.CrossRefGoogle Scholar
[12]Mints, G. E., Category theory and proof theory, Aktual'nye voprosy logiki i metodologii nauki, Naukova Dumka, Kiev, 1980, in Russian, pp. 252–278. English translation, with permuted title, in G. E. Mints, Selected papers in proof theory, Bibliopolis, Naples, 1992.Google Scholar
[13]Petrić, Z., Coherence in substructural categories, to appear, 1997.Google Scholar
[14]Soloviev, S. V., The category of finite sets and cartesian closed categories, Zapiski Nauchnykh Seminarov LOMI, vol. 105 (1981), pp. 174–194, in Russian; English translation in Journal of Soviet Mathematics, vol. 22 (1983), pp. 1387–1400.Google Scholar