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Generality of proofs and its Brauerian representation

Published online by Cambridge University Press:  12 March 2014

Kosta Došen  and
Zoran Petrić
Kosta Došen
Affiliation:
Mathematical Institute, Sanu, Knez Mihailova 35, P.F. 367, 11001 Belgrade, Yugoslavia, E-mail: kosta@mi.sanu.ac.yu
Zoran Petrić
Affiliation:
Mathematical Institute, Sanu, Knez Mihailova 35, P.F. 367, 11001 Belgrade, Yugoslavia, E-mail: zpetric@mi.sanu.ac.yu
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Abstract

The generality of a derivation is an equivalence relation on the set of occurrences of variables in its premises and conclusion such that two occurrences of the same variable are in this relation if and only if they must remain occurrences of the same variable in every generalization of the derivation. The variables in question are propositional or of another type. A generalization of the derivation consists in diversifying variables without changing the rules of inference.

This paper examines in the setting of categorial proof theory the conjecture that two derivations with the same premises and conclusions stand for the same proof if and only if they have the same generality. For that purpose generality is defined within a category whose arrows are equivalence relations on finite ordinals, where composition is rather complicated. Several examples are given of deductive systems of derivations covering fragments of logic, with the associated map into the category of equivalence relations of generality.

This category is isomorphically represented in the category whose arrows are binary relations between finite ordinals, where composition is the usual simple composition of relations. This representation is related to a classical representation result of Richard Brauer.

Keywords

identity criteria for proofsgenerality of proofcategories of proofsBrauer algebrasrepresentation
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 3 , September 2003 , pp. 740 - 750
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1] Brauer, R., On algebras which are connected with the semisimple continuous groups, Annals of Mathematics, vol. 38 (1937), pp. 857–872.CrossRefGoogle Scholar
[2] Došen, K., Identity of proofs based on normalization and generality, 2002, (available at: http://arXiv.org/math.L0/0208094).Google Scholar
[3] Došen, K., Kovijanić, Ž., and Petrić, Z., A new proof of the faithfulness of Brauer's representation of Temperley-Lieb algebras, 2002, (available at: http://arXiv.org/math.GT/0204214).Google Scholar
[4] Došen, K. and Petrić, Z., The maximality of cartesian categories. Mathematical Logic Quarterly, vol. 47 (2001), pp. 137–144, (available at: http://arXiv.org/math.CT/9911059).Google Scholar
[5] Došen, K. and Petrić, Z., Bicartesian coherence, Studia Logica, vol. 71 (2002), pp. 331–353, (available at: http://arXiv.org/math.CT/0006052).CrossRefGoogle Scholar
[6] Došen, K. and Petrić, Z., A Brauerian representation of split preorders, 2002, (available at: http://arXiv.org/math.L0/00211277).Google Scholar
[7] Eilenberg, S. and Kelly, G. M., A generalization of the functorial calculus, Journal of Algebra, vol. 3 (1966), pp. 366–375.Google Scholar
[8] Jones, V. F. R., A quotient of the affine Hecke algebra in the Brauer algebra. Enseignement des Mathématiques, vol. 40 (1994), no. 2, pp. 313–344.Google Scholar
[9] Kassel, C., Quantum groups, Springer, Berlin, 1995.CrossRefGoogle Scholar
[10] Kelly, G. M. and Lane, S. Mac, Coherence in closed categories. Journal of Pure and Applied Algebra, vol. 1 (1971), pp. 97–140 and 219.Google Scholar
[11] Lambek, J., Deductive systems and categories I: Syntactic calculus and residuated categories, Mathematical Systems Theory, vol. 2 (1968), pp. 287–318.CrossRefGoogle Scholar
[12] Lambek, J., Deductive systems and categories II: Standard constructions and closed categories, Category theory, homology theory and their applications I, Lecture Notes in Mathematics, vol. 86, Springer, Berlin, 1969, pp. 76–122.CrossRefGoogle Scholar
[13] Lambek, J., Deductive systems and categories III: Cartesian closed categories, intuitionist propositional calculus, and combinatory logic, Toposes, algebraic geometry and logic (Lawvere, F. W., editor), Lecture Notes in Mathematics, vol. 274, Springer, Berlin, 1972, pp. 57–82.Google Scholar
[14] Lambek, J., Functional completeness of cartesian categories. Annals of Mathematical Logic, vol. 6 (1974), pp. 259–292.Google Scholar
[15] Lambek, J. and Scott, P. J., Introduction to higher-order categorical logic, Cambridge University Press, Cambridge, 1986.Google Scholar
[16] Petrić, Z., Coherence in substructural categories, Studia Logica, vol. 70 (2002), pp. 271–296, (available at: http://arXiv.org/math.CT/0006061).Google Scholar
[17] Prawitz, D., Ideas and results in proof theory, Proceedings of the second Scandinavian logic symposium (Fenstad, J. E., editor), North-Holland, Amsterdam, 1971, pp. 235–307.Google Scholar
[18] Szabo, M. E., A counter-example to coherence in cartesian closed categories, Canadian Mathematical Bulletin, vol. 18 (1975), pp. 111–114.Google Scholar
[19] Wenzl, H., On the structure of Brauer's centralizer algebras, Annals of Mathematics, vol. 128 (1988). pp. 173–193.CrossRefGoogle Scholar