Advertisement

algebra universalis

, Volume 35, Issue 2, pp 265–295 | Cite as

When is a natural duality ‘good’?

  • D. M. Clark
  • B. A. Davey
Article

Abstract

Building on the most current work in the theory of natural dualities, we continue the study of strong dualities for the quasi-variety generated by a finite algebra. We investigate ten different versions of what we would like to mean by a ‘good duality’. Each version concerns, among other things, a specific restriction on the type of the structures in the dual category which insures that the dual structures will in a useful sense be simple. Through each investigation we seek a theorem characterizing, in terms of finitely verifiable conditions, those finite algebras generating a quasi-variety which admits a strong duality meeting the given restrictions. Our study includes a careful treatment of coproducts, logarithmic dualities and strong dualities by various unary structures.

1991 Mathematics Subject Classification

08C05 08C15 18A40 

Key words and phrases

Duality theory injective natural duality strong duality congruence distributivity near unanimity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Adams, M. E. andClark, D. M.,Endomorphism monoids in minimal quasi primal varieties, Acta Sci. Math. (Szeged)54 (1990), 37–52.Google Scholar
  2. [2]
    Arens, R. andKaplansky, I.,Topological representation of algebras, Trans. Amer. Math. Soc.63 (1948), 457–481.Google Scholar
  3. [3]
    Burris, S. andClark, D. M.,Elementary and algebraic properties of the Arens-Kaplansky constructions, Algebra Univ.22 (1986), 50–93.Google Scholar
  4. [4]
    Clark, D. M.,Algebraically and existentially closed Stone and double Stone algebras, Journal of Symbolic Logic54 (1989), 363–375.Google Scholar
  5. [5]
    Clark, D. M. andDavey, B. A.,The quest for strong dualities, J. Austral. Math. Soc. (Series A)58 (1995), 248–280.Google Scholar
  6. [6]
    Clark, D. M. andDavey, B. A.,Natural Dualities for the Working Algebraist, Cambridge University Press (to appear).Google Scholar
  7. [7]
    Clark, D. M. andKrauss, P. H.,Topological quasi varieties, Acta. Sci. Math. (Szeged)47 (1984), 3–39.Google Scholar
  8. [8]
    Clark, D. M. andSchmid, J.,The countable homogeneous universal model of ℬ2 (in preparation).Google Scholar
  9. [9]
    Davey, B. A.,Duality theory on ten dollars a day, Algebras and Orders (I. G. Rosenberg and G. Sabidussi, eds) NATO Advanced Study Institute Series, Series C, Vol. 389, Kluwer Academic Publishers, 1993, pp. 71–111.Google Scholar
  10. [10]
    Davey, B. A., Heindorf, L. andMcKenzie, R.,Near unanimity: an obstable to general duality theory, Algebra Univ.33 (1995), 428–439.Google Scholar
  11. [11]
    Davey, B. A. andPriestley, H. A.,Generalized piggyback dualities and applications to Ockham algebras, Houston Math. J.13 (1987), 151–197.Google Scholar
  12. [12]
    Davey, B. A., Quackenbush, R. andSchweigert, D.,Monotone clones and the varieties they determine, Order7 (1990), 145–168.Google Scholar
  13. [13]
    Davey, B. A. andWerner, H.,Dualities and equivalences for varieties of algebras, Contributions to lattice theory (Szeged, 1980), (A. P. Huhn and E. T. Schmidt, eds) Colloq. Math. Soc. János Bolyai, Vol. 33, North-Holland, Amsterdam, 1983, pp. 101–275.Google Scholar
  14. [14]
    Fleischer, I.,A note on subdirect products, Acta Math. Acad. Sci. Hungar.6 (1955), 463–465.Google Scholar
  15. [15]
    Priestley, H. A.,Ordered sets and duality for distributive lattices, Annals of Discrete Math.23 (1984), 39–60.Google Scholar
  16. [16]
    Stone, M. H.,The theory of representations for Boolean algebras, Trans. Amer. Math. Soc.40 (1936), 37–111.Google Scholar
  17. [17]
    Werner, H.,A duality for weakly associative lattices, Finite algebras and multi-valued logic, Colloq. Math. Soc. János Bolyai, Vol. 28, North-Holland, Amsterdam, 1981, pp. 781–808.Google Scholar
  18. [18]
    Werner, H.,Discriminator-algebras, Studien zur Algebra und ihre Anwendungen (Band 6), Akademie-Verlag, Berlin, 1978.Google Scholar

Copyright information

© Birkhäuser Verlag 1996

Authors and Affiliations

  • D. M. Clark
    • 1
  • B. A. Davey
    • 2
  1. 1.Mathematics and Computer ScienceSUNYCollege at New PaltzNew PaltzUSA
  2. 2.School of MathematicsLa Trobe UniversityBundooraAustralia

Personalised recommendations