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Closedness Properties of Internal Relations III: Pointed Protomodular Categories


A pointed variety of universal algebras is protomodular in the sense of D. Bourn, if and only if it is classically ideal determined in the sense of A. Ursini (this result is due to D. Bourn and G. Janelidze). We prove a characterization theorem for pointed protomodular categories, which is a (pointed) categorical version of Ursini’s characterization theorem for classically ideal determined varieties, involving classically 0-regular algebras. A suitable simplification of the property of a pair of relations, which is used to define a classically 0-regular algebra, yields a new closedness property of a single binary relation – we show that a finitely complete pointed category is protomodular if and only if every binary internal relation RA 2 in it has this closedness property.

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  1. Agliano, P., Ursini, A.: Ideals and other generalizations of congruence classes. J. Austral. Math. Soc. Ser. A 53, 103–115 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Borceux, F., Bourn, D.: Mal’cev, protomodular, homological and semi-abelian categories. In: Mathematics and its Applications, vol. 566. Kluwer, Dordrecht (2004)

    Google Scholar 

  3. Bourn, D.: Normalization equivalence, kernel equivalence and affine categories. Springer Lect. Notes Math. 1488, 43–62 (1991)

    Article  MathSciNet  Google Scholar 

  4. Bourn, D.: Mal’cev categories and fibration of pointed objects. Appl. Categor. Struct. 4, 307–327 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bourn, D.: Intrinsic centrality and associated classifying properties. J. Algebra 256, 126–145 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bourn, D., Janelidze, G.: Protomodularity, descent, and semidirect products. Theory Appl. Categ. 4(2), 37–46 (1998)

    MATH  MathSciNet  Google Scholar 

  7. Bourn, D., Janelidze, G.: Characterization of protomodular varieties of universal algebras. Theory Appl. Categ. 11(6), 143–447 (2003)

    MATH  MathSciNet  Google Scholar 

  8. Carboni, A., Lambek, J., Pedicchio, M.C.: Diagram chasing in Mal’cev categories. J. Pure Appl. Algebra 69, 271–284 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Carboni, A., Pedicchio, M.C., Pirovano, N.: Internal graphs and internal groupoids in Mal’cev categories. CMS Conf. Proc. (Category Theory 1991) 13, 97–109 (1992)

    MathSciNet  Google Scholar 

  10. Janelidze, G., Márki, L., Tholen, W.: Semi-abelian categories. J. Pure Appl. Algebra 168, 367–386 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Janelidze, Z.: Subtractive categories. Appl. Categor. Struct. 13(4), 343–350 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Janelidze, Z.: Normal product projections, subtractive categories, and elementary term conditions in universal and categorical algebra. Ph.D. thesis, A. Razmadze Mathematical Institute of Georgian Academy of Sciences (defended at I. Javakhishvili Tbilisi State University) (2006)

  13. Janelidze, Z.: Closedness properties of internal relations I: a unified approach to Mal’tsev, unital and subtractive categories. Theory Appl. Categ. 16(12), 236–261 (2006)

    MATH  MathSciNet  Google Scholar 

  14. Janelidze, Z.: Closedness properties of internal relations II: Bourn localization. Theory Appl. Categ. 16(13), 262–282 (2006)

    MATH  MathSciNet  Google Scholar 

  15. Mal’tsev, A.I.: On the general theory of algebraic systems. Mat. Sb., N.S. 35(77), 3–20 (1954) (in Russian); English translation: Amer. Math. Soc. Transl. 27(2), 125–341 (1963)

    MathSciNet  Google Scholar 

  16. Ursini, A.: Osservazioni sulla varietá BIT. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8, 205–211 (1973)

    Google Scholar 

  17. Ursini, A.: On subtractive varieties, I. Algebra Universalis 31, 204–222 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Zurab Janelidze.

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Partially supported by South African National Research Foundation, and Georgian National Science Foundation (GNSF/ST06/3-004).

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Janelidze, Z. Closedness Properties of Internal Relations III: Pointed Protomodular Categories. Appl Categor Struct 15, 325–338 (2007).

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Mathematics Subject Classifications (2000)