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Another problem of Jónsson and McKenzie from 1982: refinement properties for connected powers of posets

Abstract

It is proven in this note that if non-empty posets A, B, C, and D satisfy \(A^C\cong B^D\) where C, D, and \(A^C\) are finite and connected, then there exist posets E, X, Y, and Z such that \(A\cong E^X\), \(B\cong E^Y\), \(C\cong Y\times Z\), and \(D\cong X\times Z\). This solves a problem posed by Jónsson and McKenzie in 1982.

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References

  1. 1.

    Bauer, H.: Garben und Automorphismen geordneter Mengen. Dissertation, Technische Hochschule Darmstadt, Darmstadt, Germany (1982)

  2. 2.

    Bauer, H., Keimel, K., and Köhler, R.: Verfeinerungs- und Kürzungssätze für Produkte geordneter topologischer Räume und für Funktionen (-halb-) verbände. In: Continuous Lattices: Proceedings of the Conference on Topological and Categorical Aspects of Continuous Lattices (Workshop IV) Held at the University of Bremen, Germany, November 9–11, 1979. Lecture Notes in Mathematics, vol. 871, pp. 20–44. Springer-Verlag, Berlin (1981)

  3. 3.

    Duffus, D.: Powers of Ordered Sets. Order 1, 83–92 (1984)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Farley, J. D.: The Exponentiation Operators of Birkhoff and McKenzie, and a Counterexample Birkhoff Said in His 1942 Article He Did Not Have (2020, manuscript)

  5. 5.

    Farley, J. D.: An Issue Raised in 1978 by a Then-Future Editor-in-Chief of the Journal Order: Does the Endomorphism Poset of a Finite Connected Poset Tell Us That the Poset Is Connected? (2020, manuscript)

  6. 6.

    Jónsson, B., McKenzie, R.: Powers of Partially Ordered Sets: Cancellation and Refinement Properties. Mathematica Scandinavica 51, 87–120 (1982)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Köhler, R.: Verfeinerung bei Priestley-Räumen und Kürzungsregeln für Funktionenverbände. Diplomarbeit, Technische Hochschule Darmstadt, Darmstadt, Germany (1979)

    Google Scholar 

  8. 8.

    Kolibiar, M.: Congruence Relations and Direct Decompositions of Ordered Sets, II. In: Contributions to general algebra. 6. Dedicated to the memory of Wilfried Nöbauer. pp. 167–172. Hölder–Pichler–Tempsky, Vienna (1988)

  9. 9.

    Kolibiar, M.: (early 1990s, personal communication to the author)

  10. 10.

    McKenzie, R.: The Zig-Zag Property and Exponential Cancellation of Ordered Sets. Order 20, 185–221 (2003)

    MathSciNet  Article  Google Scholar 

  11. 11.

    McKenzie, R.N., McNulty, G. F.,and Taylor, W. F.: Algebras, Lattices, Varieties: Volume I Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, California (1987)

  12. 12.

    Nation, J. B.: Tribute to Bjarni Jónsson. Algebra Universalis 79, Paper No. 57, 13 pages (2018)

  13. 13.

    Schröder, B.S.W.: Ordered Sets, An Introduction with Connections from Combinatorics to Topology, 2nd edn. Birkhäuser Verlag, (2016)

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Acknowledgements

The author would like to thank Dr. Bernd S. W. Schröder for suggestions on improving this note. The author also thanks the referees for their comments, which he incorporated: for example, Lemma 8 and Theorem 9 originally appeared as [5, Lemma 3 and Theorem 4].

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Correspondence to Jonathan David Farley.

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Farley, J.D. Another problem of Jónsson and McKenzie from 1982: refinement properties for connected powers of posets. Algebra Univers. 82, 48 (2021). https://doi.org/10.1007/s00012-020-00698-y

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Keywords

  • (Partially) ordered set
  • Exponentiation
  • Connected

Mathematics Subject Classification

  • 06A07