Skip to main content
SpringerLink
Log in

On the structure of varieties with equationally definable principal congruences III

  • Published:
algebra universalis Aims and scope Submit manuscript

Abstract

p(x, y, z) is aternary deduction (TD) term function on an algebra A if, for alla, b ε A, p(a, b,z) ≡ z (modΘ (a, b)), and {p(a, b, z): z εA} is a transversal of the set of equivalence classes of the principal congruence θ(a, b). p iscommutative ifp(a, b, z) and p(a', b', z) define the same transversal whenever0(a, b)=0(a', b'). p isregular ifΘ(p(x, y, 1), 1)=0(x, y) for some constant term 1. The TD term generalizes the (affine) ternary discriminator and is used to investigate the logical structure of nonsemisimple varieties with equationally definable principal congruences (EDPC). Some of the results obtained: The following are equivalent for any variety: (1)V has a TD term; (2)V has EDPC and a certain strong form of the congruence-extension property. IfV is semisimple and congruence-permutable, (1) and (2) are equivalent to (3)V is an affine discriminator variety. Afixedpoint ternary discriminator on a set is defined by the conditions:p(x, x,z)=z and, ifx ≠ y, p(x, y,z)=d whered is some fixed element; afixedpoint discriminator variety is defined in analogy to affine discriminator variety. The commutative TD term generalizes the fixedpoint ternary discriminator. The following are equivalent for any semisimple variety: (4)V has a commutative TD term; (5)V is a fixedpoint discriminator variety. IfV is semisimple, congruence-permutable, and has a constant term, (4) and (5) are equivalent to (3); ifV has a second constant term distinct from the first in all nontrivial members ofV then all five conditions are equivalent to (6)V has a commutative, regular TD term. Ahoop is a commutative residuated monoid.Hoops with dual normal operators are defined in analogy with normal Boolean algebras with operators. The main result: A variety of hoops with dual normal operators has a commutative, regular TD term iff it has EDPC iff it has first-order definable principal congruences.

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. Anderson, A. R. andBelnap, N. D., Jr.,Entailment. The logic of relevance and necessity. Volume I. Princeton University Press, Princeton, 1975.

    Google Scholar 

  2. Birkhoff, G.,Lattice Theory, Amer. Math. Soc. Colloq. Publ. vol. 25, third edition, Amer. Math. Soc., New York, 1967.

    Google Scholar 

  3. Blok, W. J. andFerreirim, I. M. A.,Hoops and their implicational reducts (Abstract), Algebraic methods in logic and computer science, Banach Center Publications28 (1993), 219–230.

    Google Scholar 

  4. Blok, W. J. andPigozzi, D.,On the structure of varieties with equationally definable principal congruences I, Algebra Universalis15 (1982), 195–227.

    Google Scholar 

  5. Blok, W. J., Köhler, P. andPigozzi, D.,On the structure of varieties with equationally definable principal congruences II, Algebra Universalis18 (1984), 334–379.

    Google Scholar 

  6. Bosbach, B.,Komplementäre Halbgruppen. Ein Beitrag zur Idealtheorie kommutativer Halbgruppen, Math. Ann.161 (1965), 279–295.

    Google Scholar 

  7. Bosbach, B.,Komplementäre Halbgruppen. Axiomatik und Arithmetik, Fund. Math.64 (1969), 257–287.

    Google Scholar 

  8. Bosbach, B.,Komplementäre Halbgruppen. Kongruenzen und Quotienten, Fund. Math.69 (1970), 1–14.

    Google Scholar 

  9. Bosbach, B.,Residuation-groupoids, Bull Acad. Pol. Sci. Ser. Sci. Math. Astr. Phys.22 (1974), 103–104.

    Google Scholar 

  10. Büchi, J. R.,The collected works of J. Richard Büchi, Springer-Verlag, New York, 1990.

    Google Scholar 

  11. Büchi, J. R. andOwens, T. M.,Complemented monoids and hoops, Unpublished manuscript.

  12. Chang, C. C.,Algebraic analysis of many valued logic, Trans. Amer. Math. Soc.88 (1958), 467–490.

    Google Scholar 

  13. Cignoli, R.,Some algebraic aspects of many-valued logics, Proc. Third Brazilian Conf. Math. Logic (A. I. Arruda, N. C. A. da Costa, A. M. Sette, eds.), 1980, pp. 49–69.

  14. Cornish, W. H.,Varieties generated by finite BCK-algebras, Bull. Austral. Math. Soc.22 (1980), 411–430.

    Google Scholar 

  15. Cornish, W. H.,3-permutability and quasicommutative BCK-algebras, Math. Japonica25 (1980), 477–496.

    Google Scholar 

  16. Cornish, W. H.,BCK-algebras with a supremum, Math. Japonica27 (1982), 63–73;II. Distributivity and interpolation, Math. Japonica29 (1984), 339–347.

    Google Scholar 

  17. Curry, H. B.,Foundations of mathematical logic, Dover Publications, Inc., New York, 1977.

    Google Scholar 

  18. Diego, A.,Sur les algèbres de Hilbert, Collection de Logique Mathematique, Series A. No. 21, Gauthier-Villars, Paris, 1966.

    Google Scholar 

  19. Font, J. M. andRodríguez, G.,Note on algebraic models for relevance logic, Preprint.

  20. Font, J. M., Rodriguez, A. J. andTorrens, A.,Wajsberg algebras, Stochastica8 (1984), 5–31.

    Google Scholar 

  21. Goldblatt, R.,Topoi. The categorical analysis of logic, Studies in logic and the foundations of mathematics, Vol. 98, North-Holland, Amsterdam, 1984.

    Google Scholar 

  22. Grigolia, R.,On the algebras corresponding to the n-valued Lukasiewicz-Tarski logical systems, Proc. Fifth Inter. Symp. on Multiple-Valued Logic, Indiana University, Bloomington, Indiana 1975, IEEE, New York, 1975, pp. 234–239.

    Google Scholar 

  23. Henkin, L., Monk, J. D. andTarski, A.,Cylindric algebras, Part I, North-Holland Publishing Co., Amsterdam, 1971.

    Google Scholar 

  24. Higgs, D.,Residuated commutative monoids with identity element as least element do not form an equational class, Math. Japonica29 (1984), 69–75.

    Google Scholar 

  25. Idziak, P.,On varieties of BCK-algebras, Math. Japonica28 (1983), 157–162.

    Google Scholar 

  26. Idziak, P.,Lattice operation in BCK-algebras, Math. Japonica29 (1984), 839–846.

    Google Scholar 

  27. Idziak, P., Personal communication.

  28. Iséki, K.,An algebra related with a propositional calculus, Proc. Japan Acad.42 (1966), 26–29.

    Google Scholar 

  29. Iséki, K.,BCK-algebras with condition (S), Math. Japonica24 (1979), 107–119.

    Google Scholar 

  30. Johnstone, P. T.,Topos theory, Academic Press, London, 1977.

    Google Scholar 

  31. Jónsson, B.,Algebras whose congruence lattices are distributive, Math. Scand.21 (1967), 110–121.

    Google Scholar 

  32. Jónsson, B. andTarski, A.,Boolean algebras with operators, Parts I and II, Amer. J. Math. (1951, 1952), 891–939, 127–162.

  33. Köhler, P. andPigozzi, D.,Varieties with equationally definable principal congruences, Algebra Universalis11 (1980), 213–219.

    Google Scholar 

  34. Komori, Y.,Super-Łukasiewicz implicational logics, Nagoya Math. J.72 (1978), 127–133.

    Google Scholar 

  35. Komori, Y.,Super-Łukasiewicz propositional logics, Nagoya Math. J.84 (1981), 119–133.

    Google Scholar 

  36. Komori, Y.,The class of BCC-algebras is not a variety, Math. Japonica29 (1984), 391–394.

    Google Scholar 

  37. Komori, Y.,Predicate logics without the structure rules, Studia Logica45 (1985), 393–404.

    Google Scholar 

  38. Krull, W.,Axiomatische Begründung der allgemeinen Idealtheorie, Sitzungsberichte der physikalisch medizinischen Societät der Erlangen56 (1924), 47–63.

    Google Scholar 

  39. Lambek, J.,Deductive systems and categories I, J. Math. Systems Theory2 (1968), 278–318;II, inLecture Notes in Mathematics, Vol. 86, Springer-Verlag, New York, 1969, pp. 278–318;III, inLecture Notes in Mathematics, Vol. 274, 1972, pp. 57–82.

    Google Scholar 

  40. Lambek, J. andScott, P. J.,Introduction to higher order categorical logic, Cambridge studies in advanced mathematics 7, Cambridge University Press, Cambridge, 1986.

    Google Scholar 

  41. Lawvere, F. W.,Metric spaces, generalized logic and closed categories, Rend. Semi. Mat. Fis. Milano43 (1973), 135–166.

    Google Scholar 

  42. Łukasiewicz, J. andTarski, A.,Investigations into the sentential calculus, Logic, semantics, and metamathematics, 2nd Ed., A. Tarski, Hackett Pub. Co., Indianapolis, Indiana, 1983.

    Google Scholar 

  43. MacLane, S.,Categories for the working mathematician, Graduate texts in Mathematics 5, Springer-Verlag, New York, 1971.

    Google Scholar 

  44. Maddux, R. D.,Some varieties containing relation algebras, Trans. Amer. Math. Soc.272 (1982), 501–526.

    Google Scholar 

  45. Monteiro, A.,Cours sur les algèbres de Hilbert et de Tarski, Instituto Mat. Univ. del Sur, Bahia Bianca, 1960.

    Google Scholar 

  46. Mundici, D.,MV-algebras are categorically equivalent to bounded commutative BCK-algebras, Math. Japonica31 (1986), 889–894.

    Google Scholar 

  47. Ono, H.,Semantical analysis of predicate logics without the contraction rule, Studia Logica44 (1985), 185–196.

    Google Scholar 

  48. Ono, H.,Some remarks on semantics for the classical logic without the contraction rules, Rep. Math. Logic19 (1985), 3–12.

    Google Scholar 

  49. Ono, H. andKomori, Y.,Logics without the contraction rule, J. Symbolic Logic50 (1985), 169–201.

    Google Scholar 

  50. PaŁasiński, M.,An embedding theorem for BCK-algebras, Math. Sem. Notes Kobe Univ.10 (1982), 749–751.

    Google Scholar 

  51. PaŁasiński, M.,BCK-algebras and TD terms (to appear).

  52. Rasiowa, H.,An algebraic approach to nonclassical logics, North-Holland Publishing Co., Amsterdam, 1974.

    Google Scholar 

  53. Romanowska, A. andTraczyk, T.,On commutative BCK-algebras, Math. Japonica25 (1980), 567–583.

    Google Scholar 

  54. Romanowska, A.,Commutative BCK-algebras. Subdirectly irreducible algebras and varieties, Math. Japonica27 (1982), 35–48.

    Google Scholar 

  55. Tanaka, S.,On ∧-commutative algebras, Math. Sem. Notes Kobe Univ.3 (1975), 59–64.

    Google Scholar 

  56. Traczyk, T.,On the variety of bounded commutative BCK-algebras, Math. Japonica24 (1979), 283–292.

    Google Scholar 

  57. Ward, M. andDilworth, R. P.,Residuated lattices, Trans. Amer. Math. Soc.45 (1939), 335–354.

    Google Scholar 

  58. Werner, H.,Discriminator algebras, Studien zur Algebra und ihre Anwendungen 6, Akademie-Verlag, Berlin, 1978.

    Google Scholar 

  59. Wronski, A.,BCK-algebras do not form a variety, Math. Japonica28 (1983), 211–213.

    Google Scholar 

  60. Wronski, A.,An algebraic motivation for BCK-algebras, Math. Japonica30 (1985), 187–193.

    Google Scholar 

  61. Wronski, A.,On varieties of commutative BCK-algebras not generated by their finite members, Math. Japonica30 (1985), 227–233.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 60607, Chicago, IL, USA

    W. J. Blok & D. Pigozzi

  2. Department of Mathematics, Iowa State University, 50011, Ames, IA, USA

    W. J. Blok & D. Pigozzi

Authors
  1. W. J. Blok
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Pigozzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Bjarni Jónsson on his 70th birthday

The authors gratefully acknowledge the support of National Science Foundation Grants DMS-8703743 and DMS-8805870.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Blok, W.J., Pigozzi, D. On the structure of varieties with equationally definable principal congruences III. Algebra Universalis 32, 545–608 (1994). https://doi.org/10.1007/BF01195727

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01195727

Keywords

Search

Navigation