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Algebraic theory of quasivarieties of heterogeneous partial algebras

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Abstract

Based on existence equations, quasivarieties of heterogeneous partial algebras have the same algebraic description as those of total algebras. Because of the restriction of the valuations to the free variables of a formula — the usual reference to the needed variables e.g. for identities (in order to get useful and manageable results) is essentially replaced here by the use of the “logical Craig projections” — already varieties of heterogeneous partial algebras behave to some extent rather like quasivarieties than having the properties known from varieties of total homogeneous algebras. It is one of the main aims of this note to make this more explicit. On the other hand we want to list several results known for quasivarieties of heterogeneous partial algebras — and adopt them to the extended signature — after having recalled the language and the main concepts necessary for the understanding of the results.

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References

  1. Andréka, H., P. Burmeister, and I. Németi, ‘Quasivarieties of partial algebras — A unifying approach towards a two-valued model theory for partial algebras’, Studia Sci. Math. Hungar. 16 (1981), 325–372.

    Google Scholar 

  2. Andréka, H., and I. Németi, ‘A general axiomatizability theorem formulated in terms of cone-injective subcategories’, in B. Csákány, E. Fried, and E. T. Schmidt, (eds.), Universal Algebra, Colloq. Math. Soc. J.Bolyai, Vol. 29, North-Holland Publ. Co., Amsterdam, 1982, pp. 13–35.

    Google Scholar 

  3. Andréka, H., and I. Németi, Generalization of the concept of variety and quasivariety to partial algebras through category theory, Dissertationes Mathematicae (Rozprawy Matematyczne) No. 204, Warszawa, 1983.

  4. Binczak, G., ‘A characterization theorem for weak varieties’, Algebra Univers. 45 (2001), 53–62.

    Google Scholar 

  5. Birkhoff, G., ‘On the structure of abstract algebras’, Proc. Cambridge Philos. Soc. 31 (1935), 433–454.

    Google Scholar 

  6. Birkhoff, G., Lattice Theory, American Mathematical Society, 1967 (First edition 1941).

  7. Börner F., ‘Varieties of Partial Algebras’, Beiträge Algebra, Geom. 1996, pp. 259–287.

  8. Burmeister, P., ‘Free partial algebras’, J. reine und angewandte Math. 241 (1970), 75–86.

    Google Scholar 

  9. Burmeister, P., ‘Partial algebras — survey of a unifying approach towards a two-valued model theory for partial algebras’, Algebra Universalis 15 (1982), 306–358.

    Google Scholar 

  10. Burmeister, P., A Model Theoretic Oriented Approach to Partial Algebras. Introduction to Theory and Application of Partial Algebras — Part I., Mathematical Research Vol. 32, Akademie-Verlag, Berlin, 1986.

    Google Scholar 

  11. Burmeister, P., ‘Tools for a Theory of Partial Algebras’, in K. Denecke and H.-J. Vogel, (eds.), General Algebra and Applications, Research and Exposition in Mathematics, Vol. 20, Heldermann Verlag Berlin, 1993, pp. 12–32.

    Google Scholar 

  12. Burmeister, P., ‘Partial Algebras — An Introductory Survey’, in I. G. Rosenberg and G. Sabidussi, (eds.), Algebras and Orders, NATO ASI Series C, Vol. 389, Kluver Academic Publ., Dordrecht, London, 1993, pp. 1–70.

    Google Scholar 

  13. Burmeister, P., ‘On the equivalence of ECE-and generalized Kleene-equations for many-sorted partial algebras’, in G. Pilz, (ed.), Contributions to General Algebra 9, Verlag Hö lder-Pichler-Tempsky, Wien 1995 — Verlag B. G. Teubner, Stuttgart, pp. 91–106.

  14. Burmeister, P., ‘Subdirect representations by epimorphisms in quasivarieties of partial algebras’, in I. Chajda, G. Eigenthaler, R. Halaš, W. B. Müller, H.-J. Vogel, J. Zedník), (eds.), Contributions to General Algebra 11, Verlag Johannes Heyn, Klagenfurt, 1999, pp. 39–54.

    Google Scholar 

  15. Burmeister, P., ‘Galois Connections for Partial Algebras’, in K. Denecke, M. Erné and S. L. Wismath, (eds.), Galois Connections and Applications, Kluwer, Dordrecht, 2004, pp. 345–370.

    Google Scholar 

  16. Burmeister, P., Lecture Notes on Universal Algebra — Many Sorted Partial Algebras, Fragment of lectures hold in the summer terms of 1994, 1996, 1998, 2000 and 2002 at the Darmstadt University of Technology. Download via http://www.mathematik.tu-darmstadt.de/~burmeister/

  17. Burmeister, P., and M. Maczyński, ‘Orthomodular (partial) algebras and their representations’, Demonstratio Mathematica 27 (1994), 701–722.

    Google Scholar 

  18. Burmeister, P., R. Holzer, and M. Maczyński, ‘Quasi-rings and congruences in the theory of orthomodular algebras’, Algebra Universalis 44 (2000), 333–369.

    Google Scholar 

  19. Craig, W., ‘Near-equational and equational systems of logic for partial functions. I and II’, J. of Symb. Logic 54 (1989), 795–827 and 1181–1215.

    Google Scholar 

  20. Dvurečenskij, A., and S. Pulmannová, New Trends in Quantum Structures, Kluwer Academic Pub., 2000.

  21. Gorbunov, V. A., Algebraic Theory of Quasivarieties, Consultants Bureau, New York, 1998.

    Google Scholar 

  22. Herrlich, H., and G. E. Strecker, Category Theory — An Introduction, Allyn and Bacon, 1973, (2nd ed.: Heldermann-Verlag).

  23. Höft, H., ‘Weak and strong equations in partial algebras’, Algebra univers. 3 (1973), 203–215.

    Google Scholar 

  24. Hoehnke, H.-J., Quasi-varieties: a special access, Typescript 2002, submitted to Studia Logica.

  25. Hoehnke, H.-J., and J. Schreckenberger, Partial Algebras and their Theories, Typescript of a planned book, Bad Harzburg, 2002.

  26. Holzer, R., Programmverifikation mit partiellen Algebren als abstrakten Datentypen, Diploma thesis at the Darmstadt University of Technology, 1996. Download via http://www.mathematik.tu-damstadt.de/~holzer/

  27. Holzer, R., On subdirectly irreducible OMAs, TU-Darmstadt Preprint No. 2237, 2002; submitted to Studia Logica.

  28. Holzer, R., Describing fields by implications between strong equations, TU-Darmstadt Preprint No. 2280, 2003.

  29. Kosiuczenko, P., ‘Mal’cev type conditions for partial algebras’, Algebra Universalis 31 (1994), 467–474.

    Google Scholar 

  30. Matthiessen, G., Theorie der heterogenen Algebren, Doctoral thesis at the University of Bremen, 1976.

  31. Németi, I., and I. Sain, I., ‘Cone-implicational subcategories and some Birkhoff-type theorems’, in B. Csákány, E. Fried, and E. T. Schmidt, (eds.), Universal Algebra, Colloq. Math. Soc. J.Bolyai, Vol. 29, North-Holland Publ. Co., Amsterdam, 1982, pp. 535–578.

    Google Scholar 

  32. Robinson, A., ‘Equational logic of partial functions under Kleene equality: a complete and an incomplete set of rules’, J. Symb. Logic 54 (1989), 354–362.

    Google Scholar 

  33. Schmidt, J., ‘A general existence Theorem on partial algebras and its special cases’, Coll. Math. 14 (1966), 73–87.

    Google Scholar 

  34. Schmidt, J., ‘A homomorphism theorem for partial algebras’, Coll. Math. 21 (1970), 5–21.

    Google Scholar 

  35. Słomiński, J., ‘On the solving of systems of equations over quasi-algebras and algebras’, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 10 (1962), 627–635.

    Google Scholar 

  36. Słomiński, J., A theory of extensions of quasi-algebras to algebras, Dissertationes Mathematicae (Rozprawy Matematyczne No. 40, Warszawa, 1964.

  37. Staruch, B., and B. Staruch, ‘Strong regular varieties of partial algebras’, Algebra Univers. 31 (1994), 157–176.

    Google Scholar 

  38. Staruch, B., and B. Staruch, Algebraic characterizations of classes of partial algebras definable by strong equations and mixed strong and existential equations, Manuscript 1999, submitted to Algebra Universalis.

  39. Thiele, H., Wissenschaftstheoretische Untersuchungen in algorithmischen Sprachen I (Theorie der Graphschemata-Kalküle) VEB Deutscher Verlag der Wissenschaften, Berlin, 1966.

    Google Scholar 

  40. Wille, R., ‘Subdirekte Produkte und konjunkte Summen’, Journal f. d. reine u. angewandte Mathematik 239/240 (1970), 333–338.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Darmstadt University of Technology, Schlossgartenstr. 7, D-64289, Darmstadt, Germany

    Peter Burmeister

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  1. Peter Burmeister
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Special issue of Studia Logica: “Algebraic Theory of Quasivarieties” Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko

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Burmeister, P. Algebraic theory of quasivarieties of heterogeneous partial algebras. Stud Logica 78, 129–153 (2004). https://doi.org/10.1007/s11225-005-0153-1

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