Mathematica Bohemica

Mathematica Bohemica, Vol. 148, No. 4, pp. 435-446, 2023

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

References:
[1] G. Birkhoff: An extended arithmetic. Duke Math. J. 3 (1937), 311-316. DOI 10.1215/S0012-7094-37-00323-5 | MR 1545989 | Zbl 0016.38702
[2] G. Birkhoff: Lattice Theory. American Mathematical Society Colloquium Publications 25. AMS, New York (1948). MR 0029876 | Zbl 0033.10103
[3] B. A. Davey, H. A. Priestley: Introduction to Lattices and Order. Cambridge University Press, Cambridge (2002). DOI 10.1017/CBO9780511809088 | MR 1902334 | Zbl 1002.06001
[4] D. A. Duffus: Toward a Theory of Finite Partially Ordered Sets: Ph.D. Thesis. University of Calgary, Calgary (1978). MR 2940856
[5] D. Duffus: Automorphisms and products of ordered sets. Algebra Univers. 19 (1984), 366-369. DOI 10.1007/BF01201105 | MR 0779154 | Zbl 0554.06004
[6] D. Duffus: Powers of ordered sets. Order 1 (1984), 83-92. DOI 10.1007/BF00396275 | MR 0745591 | Zbl 0561.06005
[7] D. Duffus, I. Rival: A logarithmic property for exponents of partially ordered sets. Can. J. Math. 30 (1978), 797-807. DOI 10.4153/CJM-1978-068-x | MR 0498291 | Zbl 0497.06004
[8] D. Duffus, R. Wille: A theorem on partially ordered sets of order-preserving mappings. Proc. Am. Math. Soc. 76 (1979), 14-16. DOI 10.1090/S0002-9939-1979-0534380-2 | MR 0534380 | Zbl 0417.06001
[9] J. D. Farley: 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? Available at https://arxiv.org/abs/2005.03255v1 (2020), 12 pages.
[10] J. D. Farley: Another problem of Jónsson and McKenzie from 1982: Refinement properties for connected powers of posets. Algebra Univers. 82 (2021), Article ID 48, 6 pages. DOI 10.1007/s00012-020-00698-y | MR 4289456 | Zbl 07385383
[11] J. D. Farley: Poset exponentiation and a counterexample Birkhoff said in 1942 he did not have. Order 39 (2022), 243-250. DOI 10.1007/s11083-021-09564-5 | MR 4455054 | Zbl 07566356
[12] B. Jónsson, R. McKenzie: Powers of partially ordered sets: Cancellation and refinement properties. Math. Scand. 51 (1982), 87-120. DOI 10.7146/math.scand.a-11966 | MR 0681261 | Zbl 0501.06001
[13] L. Lovász: Operations with structures. Acta Math. Acad. Sci. Hung. 18 (1967), 321-328. DOI 10.1007/BF02280291 | MR 0214529 | Zbl 0174.01401
[14] R. McKenzie: Cardinal multiplication of structures with a reflexive relation. Fundam. Math. 70 (1971), 59-101. DOI 10.4064/fm-70-1-59-101 | MR 0280430 | Zbl 0228.08002
[15] R. McKenzie: Arithmetic of finite ordered sets: Cancellation of exponents, II. Order 17 (2000), 309-332. DOI 10.1023/A:1006449213916 | MR 1822908 | Zbl 0997.06001
[16] R. McKenzie: The zig-zag property and exponential cancellation of ordered sets. Order 20 (2003), 185-221. DOI 10.1023/B:ORDE.0000026529.04361.f8 | MR 2064045 | Zbl 1051.06003
[17] B. Schröder: Ordered Sets: An Introduction with Connections from Combinatorics to Topology. Birkhäuser, Basel (2016). DOI 10.1007/978-3-319-29788-0 | MR 3469976 | Zbl 1414.06001

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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