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

Jonathan David Farley

Mathematica Bohemica (2023)

  • Volume: 148, Issue: 4, page 435-446
  • ISSN: 0862-7959

Abstract

top
Duffus wrote in his 1978 Ph.D. thesis, “It is not obvious that P is connected and P P 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 Q .

How to cite

top

Farley, Jonathan David. "Does the endomorphism poset $P^P$ determine whether a finite poset $P$ is connected? An issue Duffus raised in 1978." Mathematica Bohemica 148.4 (2023): 435-446. <http://eudml.org/doc/299483>.

@article{Farley2023,
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 $P\cong Q$.},
author = {Farley, Jonathan David},
journal = {Mathematica Bohemica},
keywords = {(partially) ordered set; exponentiation; connected},
language = {eng},
number = {4},
pages = {435-446},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Does the endomorphism poset $P^P$ determine whether a finite poset $P$ is connected? An issue Duffus raised in 1978},
url = {http://eudml.org/doc/299483},
volume = {148},
year = {2023},
}

TY - JOUR
AU - Farley, Jonathan David
TI - Does the endomorphism poset $P^P$ determine whether a finite poset $P$ is connected? An issue Duffus raised in 1978
JO - Mathematica Bohemica
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 148
IS - 4
SP - 435
EP - 446
AB - 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 $P\cong Q$.
LA - eng
KW - (partially) ordered set; exponentiation; connected
UR - http://eudml.org/doc/299483
ER -

References

top
  1. Birkhoff, G., 10.1215/S0012-7094-37-00323-5, Duke Math. J. 3 (1937), 311-316. (1937) Zbl0016.38702MR1545989DOI10.1215/S0012-7094-37-00323-5
  2. Birkhoff, G., Lattice Theory, American Mathematical Society Colloquium Publications 25. AMS, New York (1948). (1948) Zbl0033.10103MR0029876
  3. Davey, B. A., Priestley, H. A., 10.1017/CBO9780511809088, Cambridge University Press, Cambridge (2002). (2002) Zbl1002.06001MR1902334DOI10.1017/CBO9780511809088
  4. Duffus, D. A., Toward a Theory of Finite Partially Ordered Sets: Ph.D. Thesis, University of Calgary, Calgary (1978). (1978) MR2940856
  5. Duffus, D., 10.1007/BF01201105, Algebra Univers. 19 (1984), 366-369. (1984) Zbl0554.06004MR0779154DOI10.1007/BF01201105
  6. Duffus, D., 10.1007/BF00396275, Order 1 (1984), 83-92. (1984) Zbl0561.06005MR0745591DOI10.1007/BF00396275
  7. Duffus, D., Rival, I., 10.4153/CJM-1978-068-x, Can. J. Math. 30 (1978), 797-807. (1978) Zbl0497.06004MR0498291DOI10.4153/CJM-1978-068-x
  8. Duffus, D., Wille, R., 10.1090/S0002-9939-1979-0534380-2, Proc. Am. Math. Soc. 76 (1979), 14-16. (1979) Zbl0417.06001MR0534380DOI10.1090/S0002-9939-1979-0534380-2
  9. 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?, Available at https://arxiv.org/abs/2005.03255v1 (2020), 12 pages. (2020) MR4673829
  10. Farley, J. D., 10.1007/s00012-020-00698-y, Algebra Univers. 82 (2021), Article ID 48, 6 pages. (2021) Zbl07385383MR4289456DOI10.1007/s00012-020-00698-y
  11. Farley, J. D., 10.1007/s11083-021-09564-5, Order 39 (2022), 243-250. (2022) Zbl07566356MR4455054DOI10.1007/s11083-021-09564-5
  12. Jónsson, B., McKenzie, R., 10.7146/math.scand.a-11966, Math. Scand. 51 (1982), 87-120. (1982) Zbl0501.06001MR0681261DOI10.7146/math.scand.a-11966
  13. Lovász, L., 10.1007/BF02280291, Acta Math. Acad. Sci. Hung. 18 (1967), 321-328. (1967) Zbl0174.01401MR0214529DOI10.1007/BF02280291
  14. McKenzie, R., 10.4064/fm-70-1-59-101, Fundam. Math. 70 (1971), 59-101. (1971) Zbl0228.08002MR0280430DOI10.4064/fm-70-1-59-101
  15. McKenzie, R., 10.1023/A:1006449213916, Order 17 (2000), 309-332. (2000) Zbl0997.06001MR1822908DOI10.1023/A:1006449213916
  16. McKenzie, R., 10.1023/B:ORDE.0000026529.04361.f8, Order 20 (2003), 185-221. (2003) Zbl1051.06003MR2064045DOI10.1023/B:ORDE.0000026529.04361.f8
  17. Schröder, B., 10.1007/978-3-319-29788-0, Birkhäuser, Basel (2016). (2016) Zbl1414.06001MR3469976DOI10.1007/978-3-319-29788-0

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.