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Categorical Abstract Algebraic Logic: (ℐ,N)-Algebraic Systems

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Abstract

Algebraic systems play in the theory of algebraizability of π-institutions the role that algebras play in the theory of algebraizable sentential logics. In this same sense, ℐ-algebraic systems are to a π-institution ℐ what \(\mathcal{S}\) -algebras are to a sentential logic \(\mathcal{S}\) . More precisely, an (ℐ,N)-algebraic system is the sentence functor reduct of an N′-reduced (N,N′)-full model of a π-institution ℐ. Algebraic systems are formally introduced and their relationship with full models and with bilogical morphisms is investigated.

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References

  1. Barr, M. and Wells, C.: Category Theory for Computing Science, 3rd edn, Les Publications CRM, Montréal, 1999.

    Google Scholar 

  2. Blok, W. J. and Pigozzi, D.: Protoalgebraic logics, Studia Logica 45 (1986), 337–369.

    Google Scholar 

  3. Blok, W. J. and Pigozzi, D.: Algebraizable logics, Mem. Amer. Math. Soc. 77(396) (1989).

  4. Blok, W. J. and Pigozzi, D.: Algebraic semantics for universal Horn logic without equality, in A. Romanowska and J. D. H. Smith (eds.), Universal Algebra and Quasigroup Theory, Heldermann Verlag, Berlin, 1992.

    Google Scholar 

  5. Borceux, F.: Handbook of Categorical Algebra, Encyclopedia Math. Appl. 50, Cambridge University Press, Cambridge, U.K., 1994.

    Google Scholar 

  6. Czelakowski, J.: Equivalential logics I, II, Studia Logica 40 (1981), 227–236, 355–372.

    Google Scholar 

  7. Czelakowski, J.: Protoalgebraic Logics, Studia Logica Library 10, Kluwer, Dordrecht, 2001.

    Google Scholar 

  8. Diaconescu, R.: Grothendieck institutions, Appl. Categ. Structures 10(4) (2002), 383–402.

    Google Scholar 

  9. Diaconescu, R.: Institution-independent ultraproducts, Fund. Inform. 55(3–4) (2002), 321–348.

    Google Scholar 

  10. Diaconescu, R.: An institution-independent proof of the Craig interpolation property, Studia Logica 77(1) (2004), 59–79.

    Google Scholar 

  11. Fiadeiro, J. and Sernadas, A.: Structuring theories on consequence, in D. Sannella and A. Tarlecki (eds.), Recent Trends in Data Type Specification, Lecture Notes in Comput. Sci. 332, Springer-Verlag, New York, 1988, pp. 44–72.

    Google Scholar 

  12. Font, J. M. and Jansana, R.: A General Algebraic Semantics for Sentential Logics, Lecture Notes in Logic 7, Springer-Verlag, Berlin, 1996.

    Google Scholar 

  13. Font, J. M., Jansana, R. and Pigozzi, D.: A survey of abstract algebraic logic, Studia Logica 74(1/2) (2003), 13–97.

    Google Scholar 

  14. Goguen, J. A. and Burstall, R. M.: Introducing institutions, in E. Clarke and D. Kozen (eds.), Proceedings of the Logic of Programming Workshop, Lecture Notes in Comput. Sci. 164, Springer-Verlag, New York, 1984, pp. 221–256.

    Google Scholar 

  15. Goguen, J. A. and Burstall, R. M.: Institutions: Abstract model theory for specification and programming, J. Assoc. Comput. Mach. 39(1) (1992), 95–146.

    Google Scholar 

  16. Herrmann, B.: Equivalential logics and definability of truth, Dissertation, Freie Universitat Berlin, Berlin, 1993.

  17. Herrmann, B.: Equivalential and algebraizable logics, Studia Logica 57 (1996), 419–436.

    Google Scholar 

  18. Herrmann, B.: Characterizing equivalential and algebraizable logics by the Leibniz operator, Studia Logica 58 (1997), 305–323.

    Google Scholar 

  19. Mac Lane, S.: Categories for the Working Mathematician, Springer-Verlag, 1971.

  20. Prucnal, T. and Wroński, A.: An algebraic characterization of the notion of structural completeness, Bull. of the Section of Logic 3 (1974), 30–33.

    Google Scholar 

  21. Voutsadakis, G.: Categorical abstract algebraic logic, Doctoral Dissertation, Iowa State University, Ames, Iowa, 1998.

  22. Voutsadakis, G.: Categorical abstract algebraic logic: Algebraizable institutions, Appl. Categ. Structures 10(6) (2002), 531–568.

    Google Scholar 

  23. Voutsadakis, G.: Categorical abstract algebraic logic: Equivalent institutions, Studia Logica 74(1/2) (2003), 275–311.

    Google Scholar 

  24. Voutsadakis, G.: A categorical construction of a variety of clone algebras, Sci. Math. Japon. 8 (2003), 215–225.

    Google Scholar 

  25. Voutsadakis, G.: On the categorical algebras of first-order logic, Sci. Math. Japon. 10 (2004), 47–54.

    Google Scholar 

  26. Voutsadakis, G.: Categorical abstract algebraic logic: Categorical algebraization of equational logic, Logic J. IGPL 12(4) (2004), 313–333.

    Google Scholar 

  27. Voutsadakis, G.: Categorical abstract algebraic logic: Categorical algebraization of first-order logic without terms, To appear in the Arch. Math. Logic, Preprint available at http://pigozzi.lssu.edu/WWW/research/papers.html

  28. Voutsadakis, G.: Categorical abstract algebraic logic: Tarski congruence systems, logical morphisms and logical quotients, Submitted to the Ann. Pure Appl. Logic, Preprint available at http://pigozzi.lssu.edu/WWW/research/papers.html

  29. Voutsadakis, G.: Categorical abstract algebraic logic: Models of π-institutions, To appear in the Notre Dame J. Formal Logic, Preprint available at http://pigozzi.lssu.edu/WWW/research/papers.html

  30. Voutsadakis, G.: Categorical abstract algebraic logic: Generalized Tarski congruence systems, Submitted to Theory and Applications of Categories, Preprint available at http://pigozzi.lssu.edu/WWW/research/papers.html

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  1. School of Mathematics and Computer Science, Lake Superior State University, 650 W. Easterday Avenue, Sault Sainte Marie, MI, 49783, U.S.A.

    George Voutsadakis

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  1. George Voutsadakis
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Correspondence to George Voutsadakis.

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Mathematics Subject Classifications (2000)

Primary: 03Gxx, secondary: 18Axx, 68N05.

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Voutsadakis, G. Categorical Abstract Algebraic Logic: (ℐ,N)-Algebraic Systems. Appl Categor Struct 13, 265–280 (2005). https://doi.org/10.1007/s10485-005-5797-5

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