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


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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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).

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  • (Partially) ordered set
  • Exponentiation
  • Connected

Mathematics Subject Classification

  • 06A07