Mathematica Bohemica

Mathematica Bohemica, first online, pp. 1-12

Does the endomorphism poset $P^P$ determine whether a finite poset $P$ is connected? An issue Duffus raised in 1978

Jonathan David Farley

Received January 19, 2022. Published online August 29, 2022.

Abstract: Duffus wrote in his 1978 Ph.D. thesis, "It is not obvious that $P$ is connected and $P^P\cong Q^Q$ imply that $Q$ is connected", where $P$ and $Q$ are finite nonempty posets. We show that, indeed, under these hypotheses $Q$ is connected and $P\cong Q$.

Keywords: (partially) ordered set; exponentiation; connected

Classification MSC: 06A07

DOI: 10.21136/MB.2022.0010-22

PDF available at: Institute of Mathematics CAS

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Affiliations: Jonathan David Farley, Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, United States of America, e-mail: lattice.theory@gmail.com

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