The shuffle Hopf algebra and noncommutative full completeness

Journal of Symbolic Logic

Volumn 63, Issue 4, 1998, Pages 1413-1436

  Blute, R F ^a   Scott, P J ^a  

^a UNIVERSITY OF OTTAWA   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032275449     PISSN: 00224812     EISSN: None     Source Type: Journal    
DOI: 10.2307/2586659     Document Type: Article

References (52)

1. K. ABÉ, Hopf algebras, Cambridge University Press, 1977.
2. S. ABRAMSKY, Semantics of interaction: an introduction to game semantics, in A. Pitts and P. Dybjer [40], pp. 1-31.
3. S. ABRAMSKY and R. JAGADEESAN, Games and full completeness for multiplicative linear logic, this JOURNAL, vol. 59 (1994), no. 2, pp. 543-574.
4. V. M. ABRUSCI, Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic, this JOURNAL, vol. 56 (1991), pp. 1403-1456.
5. E. BAINBRIDGE, P. FREYD, A. SCEDROV, and P. SCOTT, Functorial polymorphism, Theoretical Computer Science, vol. 70 (1990), pp. 1403-1456.
6. M. BARR, Duality of Banach spaces, Cahiers de Topologie et Géométrie Différentielle Catégoriques, vol. 17 (1976), pp. 15-32.
7. _, Duality of vector spaces, Cahiers de Topologie et Géométrie Différentielle Catégoriques, vol. 17 (1976), pp. 3-14.
8. _, *-autonomous categories, Lecture Notes in Mathematics, no. 752, Springer-Verlag, 1980.
9. _, Noncommutative *-autonomous categories, preprint, 1993.
10. _, Separability of tensor in Chu categories of vector spaces, Mathematical Structures in Computer Science, vol. 6 (1996), pp. 213-217.
11. D. B. BENSON, Bialgebras: some foundations for distributed and concurrent computation, Fundamenta Informaticae, vol. 12 (1989), pp. 427-486.
12. R. BLUTE, Linear logic, coherence and dinaturality, Theoretical Computer Science, vol. 115 (1993), pp. 3-41.
13. _, Hopf algebras and linear logic, Mathematical Structures in Computer Science, vol. 6 (1996), pp. 189-212.
14. R. F. BLUTE and P. J. SCOTT, Linear Läuchli semantics, Annals of Pure and Applied Logic, vol. 77 (1996), pp. 101-142.
15. _, A noncommutative full completeness theorem (extended abstract), Electronic Notes in Theoretical Computer Science, vol. 3 (1996), Elsevier Science B. V.
16. J. R. B. COCKETT and R. A. G. SEELY, Proof theory for full intuitionistic linear logic, bilinear logic and mix categories, Theory and Applications of Categories, vol. 3 (1997), pp. 85-131.
17. T. COQUAND, C. GUNTER, and G. WINSKEL, Domain-theoretic models of polymorphism, Information and Computation, vol. 81 (1989), no. 2, pp. 123-167.
18. V. DANOS, La logique linéaire appliquée à l'étude de divers processus de normalisation et principalement du λ-calcul, Thèse du doctorat, Université de Paris, 1990.
19. V. DANOS and L. REGNIER, The structure of multiplicatives, Archive for Mathematical Logic, vol. 28 (1989), pp. 181-203.
20. A. FLEURY, La règle d'échange: logique linéaire multiplicative tressée Thèse du doctorat, U. Paris VII, 1996.
21. A. FLEURY and C. RÉTORÉ, The mix rule, Mathematical Structures in Computer Science, vol. 4 (1994), pp. 273-285.
22. J. Y. GIRARD, The system F of variable types, fifteen years later, Theoretical Computer Science, vol. 45 (1986), no. 2, pp. 159-192.
23. _, Linear logic, Theoretical Computer Science, vol. 50 (1987), pp. 1-102.
24. _, Coherent Banach spaces, preprint and lectures delivered at Keio University, Linear Logic '96, April 1996, 1996.
25. J. Y. GIRARD, Y. LAFONT, and P. TAYLOR, Proofs and types, Cambridge Tracts in Theoretical Computer Science, no. 7, 1989.
26. J. Y. GIRARD, A. SCEDROV, and P. SCOTT, Normal forms and cut-free proofs as natural transformations, Logic from computer science, Mathematical Science Research Institute Publications, no. 21, 1991. also available from http://hypatia.dcs.qmw.ac.uk. under Scott, pp. 217-241.
27. V. HARNIK and M. MAKKAI, Lambek's categorical proof theory and Läuchli's abstract realizability, this JOURNAL, vol. 57 (1992), pp. 200-230.
28. M. HAZEWINKEL, Introductory recommendations for the study of Hopf algebras in mathematics and physics, CWI Quarterly, vol. 4 (1991), no. 1, Centre tor Mathematics and Computer Science, Amsterdam.
29. J. M. E. HYLAND, Game semantics, in A. Pitts and P. Dybjer [40], pp. 131-184.
30. J. M. E. HYLAND and C.-H. L. ONG, On full abstraction for PCF, to appear, 1995.
31. S. JONI and G. C. ROTA, Coalgebras and bialgebras in combinatorics, Studies in Applied Mathematics, vol. 61 (1979), pp. 93-139.
32. A. JOYAL, Une théorie combinatoire des séries formelles, Advances in Mathematics, vol. 42 (1981), pp. 1-82.
33. J. LAMBEK, Bilinear logic in algebra and linguistics. Advances in linear logic, London Mathematical Society Series, no. 222, Cambridge University Press, Cambridge, 1995.
34. H. LÄUCHLI, An abstract notion of realizability for which intuitionistic predicate calculus is complete, Intuitionism and proof theory, North-Holland, 1970, pp. 227-234.
35. F. W. LAWVERE, Equality in hyperdoctrines and comprehension schema as an adjoint functor, Proceedings of the New York symposium on applications of categorical algebra (A. Heller, editor), American Mathematical Society, pp. 1-14.
36. _, Adjointness in foundations, Dialectica, vol. 23 (1969), pp. 281-296.
37. S. LEFSCHETZ, Algebraic topology, American Mathematical Society Colloquium Publications, no. 27, 1963.
38. S. MAJID, Physics for algebraists: noncommutative and noncocommutative Hopf algebras by a bicrossproduct construction, Journal of Algebra, vol. 130 (1990), pp. 17-64.
39. _, Quasitriangular Hopf algebras and Yang-Baxter equations, International Journal of Modern Physics, vol. 5 (1990), pp. 1-91.
40. A. Pitts and P. Dybjer (editors), Semantics and logics of computation, Publications of the Newton Institute, Cambridge University Press, 1997.
41. G. PLOTKIN, Lambda definability in the full type hierarchy, To H. B. Curry, essays on combinatory logic, Lambda calculus, and formalism, Academic Press, 1980, pp. 363-373.
42. _, August 1996, private communication.
43. G. PLOTKIN and M. ABADI, A logic for parametric polymorphism, Typed Lambda calculus and applications, Lecture Notes in Computer Science, no. 664, Springer-Verlag, 1993, pp. 361-375.
44. C. RETORÉ, Calcul de Lambek et logique linéaire, Traitement Automatique des Langues, vol. 37, no. 2, pp. 39-70.
45. C. RETORÉ Des réseaux de démonstration pour la linguistique, manuscript, 1996.
46. K. I. ROSENTHAL, *-autonomous categories of bimodules, Journal of Pure and Applied Algebra, vol. 97 (1994), pp. 189-202.
47. W. SCHMITT, Antipodes and incidence coalgebras, Journal of Combinatorial Theory, vol. 46 (1987), pp. 264-290.
48. _, Hopf algebras of combinatorial structures, Canadian Journal of Mathematics, vol. 45 (1993), pp. 412-428.
49. M. SWEEDLER, Hopf algebras, Benjamin Press, 1969.
50. F. TREVES, Topological vector spaces, distributions and kernels, Academic Press, 1967.
51. K. ULBRICH, On Hopf algebras and rigid monoidal categories, Israel Journal of Mathematics, vol. 71 (1989), pp. 252-256.
52. D. YETTER, Quantales and (noncommutative) linear logic, this JOURNAL, vol. 55 (1990), pp. 41-64.