Skip to main content
SpringerLink
Log in

Categorical Abstract Algebraic Logic: Leibniz Equality and Homomorphism Theorems

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

The study of structure systems, an abstraction of the concept of first-order structures, is continued. Structure systems have algebraic systems rather than universal algebras as their algebraic reducts. Moreover, their relational component consists of a collection of relation systems on the underlying functors rather than simply a system of relations on a single set. Congruence systems of structure systems are introduced and the Leibniz congruence system of a structure system is defined. Analogs of the Homomorphism, the Second Isomorphism and the Correspondence Theorems of Universal Algebra are provided in this more abstract context. These results generalize corresponding results of Elgueta for equality-free first-order logic. Finally, a version of Gödel’s Completeness Theorem is provided with reference to structure systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    MATH  Google Scholar 

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

  3. Bloom, S.L.: Some theorems on structural consequence operations. Studia Logica 34, 1–9 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  4. Borceux, F.: Handbook of Categorical Algebra. Encyclopedia of Math. Appl. vol. 50. Cambridge University Press, Cambridge, UK (1994)

    Google Scholar 

  5. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, Berlin Heidelberg New York (1981)

    MATH  Google Scholar 

  6. Casanovas, E., Dellunde, P., Jansana, R.: On elementary equivalence for equality-free logic. Notre Dame J. Formal Logic 37(3), 506–522 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chang, C.C., Keisler, H.J.: Model Theory. Elsevier, Amsterdam (1990)

    Google Scholar 

  8. Czelakowski, J.: Protoalgebraic Logics. Kluwer, Dordtrecht (2001)

    MATH  Google Scholar 

  9. Czelakowski, J., Elgueta, R.: Local characterization theorems for some classes of structures. Acta Sci. Math. 65, 19–32 (1999)

    MATH  MathSciNet  Google Scholar 

  10. Dellunde, P.: Equality-free logic: the method of diagrams and preservation theorems. Log. J. IGPL 7(6), 717–732 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dellunde, P.: On definability of the equality in classes of algebras with an equivalence relation. Studia Logica 64, 345–353 (2000)

    Article  MathSciNet  Google Scholar 

  12. Dellunde, P., Jansana, R.: Some characterization theorems for infinitary universal horn logic without equality. J. Symbolic Logic 61, 1242–1260 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Doets, K.: Basic Model Theory. CSLI Publications, Stanford, California (1996)

    MATH  Google Scholar 

  14. Elgueta, R.: Characterizing classes defined without equality. Studia Logica 58, 357–394 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Elgueta, R.: Subdirect representation theory for classes without equality. Algebra Universalis 40, 201–246 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Elgueta, R.: Algebraic characterizations for universal fragments of logic. Math. Logic Quart. 45, 385–398 (1999)

    MATH  MathSciNet  Google Scholar 

  17. Elgueta, R.: Freeness in classes without equality. J. Symbolic Logic 64(3), 1159–1194 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Elgueta, R., Jansana, R.: Definability of Leibniz equality. Studia Logica 63, 223–243 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Font, J.M., Jansana, R.: A General Algebraic Semantics for Sentential Logics. Lecture Notes Logic, vol. 7. Springer, Berlin Heidelberg New York (1996)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Hodges, W.: Model Theory. Cambridge University Press, UK (1993)

    MATH  Google Scholar 

  22. Mac Lane, S.: Categories for the Working Mathematician. Springer, Berlin Heidelberg New York (1971)

    MATH  Google Scholar 

  23. Marker, D.: Model Theory: An Introduction. Springer, Berlin Heidelberg New York (2002)

    Google Scholar 

  24. McKenzie, R.N., McNulty, G.F., Taylor, W.F.: Algebras, Lattices, Varieties, vol. I. Wadsworth & Brooks/Cole, Monterey, California (1987)

    Google Scholar 

  25. Pigozzi, D.: Partially Ordered Varieties and QuasiVarieties. (2004) Preprint available at http://www.math.iastate.edu/dpigozzi/

  26. 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://www.voutsadakis.com/RESEARCH/papers.html

  27. Voutsadakis, G.: Categorical abstract algebraic logic: models of π-institutions. Notre Dame J. Formal Logic 46(4), 439–460 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Voutsadakis, G.: Categorical abstract algebraic logic: \((\mathcal{I},N)\)-algebraic systems. Appl. Categ. Structures 13(3), 265–280 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  29. Voutsadakis, G.: Categorical abstract algebraic logic: prealgebraicity and protoalgebraicity. To appear in Studia Logica Preprint available at http://www.voutsadakis.com/RESEARCH/papers.html

  30. Voutsadakis, G.: Categorical abstract algebraic logic: partially ordered algebraic systems. Appl. Categ. Structures 14(1), 81–98 (2006)

    Article  MathSciNet  Google Scholar 

  31. Voutsadakis, G.: Categorical abstract algebraic logic: structure systems and łos’ theorem. To appear in the Far East J. Math. Sci. Preprint available at http://www.voutsadakis.com/RESEARCH/papers.html

Download references

Author information

Authors and Affiliations

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

    George Voutsadakis

Authors
  1. George Voutsadakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George Voutsadakis.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Voutsadakis, G. Categorical Abstract Algebraic Logic: Leibniz Equality and Homomorphism Theorems. Appl Categor Struct 14, 357–376 (2006). https://doi.org/10.1007/s10485-006-9030-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-006-9030-y

Key words

Mathematics Subject Classifications (2000)

Search

Navigation