A 2-categorical framework for the syntax and semantics of many-sorted equational logic

• Autorzy: Vidal C. , Soliveres Tur J.
• Języki publikacji: PL

Abstrakty

EN

For, not necessarily similar, single-sorted algebras Fujiwara defined, through the concept of family of basic mappingformulas between single-sorted signatures, a notion of morphism which generalizes the ordinary notion of homomorphism between algebras. Subsequently he also defined an equivalence relation, the relation of conjugation, on the families of basic mapping-formulas. In this article we extend the theory of Fujiwara to the, not necessarily similar, many-sorted algebras, by defining the concept of polyderivor between many-sorted signatures under which are subsumed the standard signature morphisms, the derivors of Goguen- Thatcher-Wagner, and the basic mapping-formulas of Fujiwara.

Słowa kluczowe

EN

Many-sorted set; Many-sorted algebra; Ehresmann-Grothendieck construction; Kleisli category for a monad; many-sorted term; clones; many-sorted algebraic theory; Hall algebra; B´enabou algebra; polyderivor; transformation of polyderivors

• Wydawca: Wydawnictwo Uniwersytetu Jagiellońskiego
• Czasopismo: Reports on Mathematical Logic
• Rocznik: 2010
• Tom: Vol. 45
• Strony: 37--95
• Opis fizyczny: Bibliogr. 25 poz.

Twórcy

autor

Vidal C.

autor

Soliveres Tur J.

• Departamento de Logica y Filosofia de la Ciencia E-46010 Valencia, Spain

Bibliografia

• [1] J. Benabou, Structures algebriques dans les categories, Cahiers de topologie et geometrie diff´erentielle 10 (1968), pp. 1–126.
• [2] G. Birkhoff and J. D. Lipson, Heterogeneous algebras, J. Combinatorial Theory 8(1970), pp. 115–133.
• [3] F. Borceux and G. Janelidze, Galois theories, Cambridge University Press. Cambridge Studies in Advanced Mathematics 72. Cambridge, 2001.
• [4] J. Climent and J. Soliveres, The completeness theorem for monads in categories of sorted sets, Houston Journal of Mathematics 31 (2005), pp. 103–129.
• [5] J. Climent and J. Soliveres, On the completeness theorem of many-sorted equational logic and the equivalence between Hall algebras and B´enabou theories, Reports on Mathematical Logic 40 (2006), pp. 127–158.
• [6] P. Cohn, Universal algebra, D. Reidel Publishing Company, Dordrecht: Holland / Boston: U.S.A. / London: England, 1981.
• [7] Ch. Ehresmann, Cat´egories et structures, Dunod, Paris, 1965.
• [8] T. Fujiwara, On mappings between algebraic systems, Osaka Math. J., 11 (1959),pp. 153–172.
• [9] T. Fujiwara, On mappings between algebraic systems, II, Osaka Math. J. 12 (1960),pp. 253–268.
• [10] K. G¨odel, Eine interpretation des intuitionistischen Aussagenkalk¨uls, Ergebnisse eines mathematischen Kolloquiums 4 (1933), pp. 39–40.
• [11] J. Goguen and R. Burstall, Introducing institutions. In E. Clarke and D. Kozen,editors, Logics of programs. Proceedings of the fourth workshop held at Carnegie-Mellon University (Pittsburgh, Pa., June 6–8, 1983). Springer-Verlag, Berlin, 1984,pp. 221–256.
• [12] J. Goguen and J. Meseguer, Completeness of many-sorted equational logic, Houston Journal of Mathematics 11 (1985), pp. 307–334.
• [13] J. Goguen, J. Thatcher and E. Wagner, An initial algebra approach to the specification, correctness, and implementation of abstract data types, IBM Thomas J.Watson Research Center, Tecnical Report RC 6487, October 1976.
• [14] G. Grätzer, Universal algebra, 2nd ed., Springer-Verlag, New York · Heidelberg ·Berlin, 1979.
• [15] A. Grothendieck, Categories fibrees et descente (Expose VI). In A. Grothendieck,editor, Revetements Etales et Groupe Fondamental (SGA 1), Springer-Verlag, Berlin· Heidelberg · New York, 1971, pp. 145–194.
• [16] P.J. Higgins, Algebras with a scheme of operators, Mathematische Nachrichten 27(1963), pp. 115–132.
• [17] N. Jacobson, Structure of rings. Amer. Math. Soc. Colloq. Pub., vol. 37. Revised edition. American Math. Soc., Providence, R. I., 1968.
• [18] H. Kleisli, Every standard construction is induced by a pair of adjoint functors, Proc. Amer. Math. Soc. 16 (1965), pp. 544–546.
• [19] F. W. Lawvere, Functorial semantics of algebraic theories, Dissertation. Columbia University, 1963.
• [20] M. Lazard, Lois de groupes et analyseurs, Ann. Sci. Ecole Norm. Sup. 72 (1955), pp. 299–400.
• [21] S. Mac Lane, Categories for the working mathematician. 2nd ed., Springer-Verlag,New York · Berlin · Heidelberg, 1998.
• [22] G. Matthiessen, Theorie der Heterogenen Algebren, Mathematik-Arbeits-Papiere,Nr. 3, Universit¨at, Bremen, 1976.
• [23] E. Post. Introduction to a general theory of elementary propositions, Amer. J. Math. 43 (1921), pp. 163–185.
• [24] E. Post, The two-valued iterative systems of mathematical logic. Annals of Math. Studies, no. 5., Princeton Univ. Press, Princeton, N. J., 1941.
• [25] A. Tarlecki, R. Burstall and J. Goguen, Some fundamental algebraic tools for the semantics of computation: Part 3. Indexed categories, Theoretical Computer Science 91 (1991), pp. 239–264.