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Counting relations on Ockham algebras

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Abstract

We find all finite Ockham algebras that admit only finitely many compatible relations (modulo a natural equivalence). Up to isomorphism and symmetry, these Ockham algebras form two countably infinite families: one family consists of the quasi-primal Ockham algebras, and the other family is a sequence of generalised Stone algebras.

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References

  1. Baker K.A., Pixley A.F.: Polynomial interpolation and the Chinese remainder theorem for algebraic systems. Math. Z. 143, 165–174 (1975)

    Article  MathSciNet  Google Scholar 

  2. Berman J.: Distributive lattices with an additional unary operation. Aequationes Math. 16, 165–171 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blyth T.S., Varlet J.C.: On a common abstraction of De Morgan algebras and Stone algebras. Proc. Roy. Soc. Edinburgh Sect. A 94, 301–308 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blyth T.S., Varlet J.C.: On the dual space of an MS-algebra. Math. Pannon. 1, 95–109 (1990)

    MathSciNet  MATH  Google Scholar 

  5. Blyth T.S., Varlet J.C.: Ockham Algebras. Oxford University Press, New York (1994)

    MATH  Google Scholar 

  6. Clark D.M., Davey B.A.: Natural Dualities for the Working Algebraist. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  7. Clark, D.M., Davey, B.A., Freese, R.S., Jackson, M.: Standard topological algebras: syntactic and principal congruences and profiniteness. Algebra Universalis 52, 343–376 (2004)

  8. Cornish W.H., Fowler P.R.: Coproducts of De Morgan algebras. Bull. Austral. Math. Soc. 16, 1–13 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  9. Davey, B.A.: Topological duality for prevarieties of universal algebras. In: Rota, G.-C. (ed.) Studies in Foundations and Combinatorics. Adv. in Math. Suppl. Stud., vol. 1, pp. 61–99. Academic Press, New York (1978)

  10. Davey B.A.: Dualities for Stone algebras, double Stone algebras, and relative Stone algebras. Colloq. Math. 46, 1–14 (1982)

    MathSciNet  MATH  Google Scholar 

  11. Davey B.A., Haviar M.: Applications of Priestley duality in transferring optimal dualities. Studia Logica 78, 213–236 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Davey B.A., Pitkethly J.G.: Counting the relations compatible with an algebra. Internat. J. Algebra Comput. 20, 901–922 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Davey B.A., Pitkethly J.G., Willard R.: The lattice of alter egos. Internat. J. Algebra Comput. 22, 1250007 (2012)

    Article  MathSciNet  Google Scholar 

  14. Davey B.A., Priestley H.A.: Generalised piggyback dualities and applications to Ockham algebras. Houston J. Math. 13, 151–198 (1987)

    MathSciNet  MATH  Google Scholar 

  15. Davey B.A., Priestley H.A.: Optimal natural dualities. Trans. Amer. Math. Soc. 338, 655–677 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Davey B.A., Priestley H.A.: Optimal natural dualities for varieties of Heyting algebras. Studia Logica 56, 67–96 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Davey B.A., Werner H.: Piggyback-Dualitäten. Bull. Austral. Math. Soc. 32, 1–32 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  18. Davey, B.A., Werner, H.: Piggyback dualities. In: Szabó, L. and Szendrei, Á. (eds.) Lectures in Universal Algebra (Szeged, 1983). Colloq. Math. Soc. János Bolyai, vol. 43, pp. 61–83. North-Holland, Amsterdam (1986)

  19. Goldberg M.S.: Distributive Ockham algebras: free algebras and injectivity. Bull. Austral. Math. Soc. 24, 161–203 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  20. Goldberg M.S.: Topological duality for distributive Ockham algebras. Studia Logica 42, 23–31 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nguyen L.T., Pitkethly J.G.: Compatible relations on Heyting chains. Algebra Universalis 67, 63–80 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Priestley H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2, 186–190 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  23. Priestley H.A.: Ordered topological spaces and the representation of distributive lattices. Proc. London Math. Soc. (3) 24, 507–530 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  24. Urquhart A.: Distributive lattices with a dual homomorphic operation. Studia Logica 38, 201–209 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  25. Urquhart A.: Distributive lattices with a dual homomorphic operation. II. Studia Logica 40, 391–404 (1981)

    Article  MathSciNet  MATH  Google Scholar 

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Author notes

  1. Long T. Nguyen

    Present address: Quang Tri Teacher Training College, Dong Ha City, Quang Tri Province, Vietnam

Authors and Affiliations

  1. Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia

    Brian A. Davey, Long T. Nguyen & Jane G. Pitkethly

Authors
  1. Brian A. Davey
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  2. Long T. Nguyen
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  3. Jane G. Pitkethly
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Corresponding author

Correspondence to Brian A. Davey.

Additional information

Presented by M. Jackson.

Dedicated with best wishes from the second and third authors to the first author, Brian Davey, on the occasion of his 65th birthday.

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Davey, B.A., Nguyen, L.T. & Pitkethly, J.G. Counting relations on Ockham algebras. Algebra Univers. 74, 35–63 (2015). https://doi.org/10.1007/s00012-015-0334-3

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