Does the endomorphism poset $P^P$ determine whether a finite poset $P$ is connected? An issue Duffus raised in 1978
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 $P\cong Q$.
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| Format: | Article |
| Language: | English |
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Institute of Mathematics of the Czech Academy of Science
2023-12-01
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| Series: | Mathematica Bohemica |
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| Online Access: | http://mb.math.cas.cz/full/148/4/mb148_4_1.pdf |
| _version_ | 1797540027799961600 |
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| author | Jonathan David Farley |
| author_facet | Jonathan David Farley |
| author_sort | Jonathan David Farley |
| collection | DOAJ |
| description | 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 $P\cong Q$. |
| first_indexed | 2024-03-10T12:55:09Z |
| format | Article |
| id | doaj.art-6e8a5dd7f1e44a7f9d0b4f005a874e3d |
| institution | Directory Open Access Journal |
| issn | 0862-7959 2464-7136 |
| language | English |
| last_indexed | 2024-03-10T12:55:09Z |
| publishDate | 2023-12-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj.art-6e8a5dd7f1e44a7f9d0b4f005a874e3d2023-11-21T12:00:13ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362023-12-01148443544610.21136/MB.2022.0010-22MB.2022.0010-22Does the endomorphism poset $P^P$ determine whether a finite poset $P$ is connected? An issue Duffus raised in 1978Jonathan David FarleyDuffus 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 $P\cong Q$.http://mb.math.cas.cz/full/148/4/mb148_4_1.pdf (partially) ordered set exponentiation connected |
| spellingShingle | Jonathan David Farley Does the endomorphism poset $P^P$ determine whether a finite poset $P$ is connected? An issue Duffus raised in 1978 Mathematica Bohemica (partially) ordered set exponentiation connected |
| title | Does the endomorphism poset $P^P$ determine whether a finite poset $P$ is connected?
An issue Duffus raised in 1978 |
| title_full | Does the endomorphism poset $P^P$ determine whether a finite poset $P$ is connected?
An issue Duffus raised in 1978 |
| title_fullStr | Does the endomorphism poset $P^P$ determine whether a finite poset $P$ is connected?
An issue Duffus raised in 1978 |
| title_full_unstemmed | Does the endomorphism poset $P^P$ determine whether a finite poset $P$ is connected?
An issue Duffus raised in 1978 |
| title_short | Does the endomorphism poset $P^P$ determine whether a finite poset $P$ is connected?
An issue Duffus raised in 1978 |
| title_sort | does the endomorphism poset p p determine whether a finite poset p is connected an issue duffus raised in 1978 |
| topic | (partially) ordered set exponentiation connected |
| url | http://mb.math.cas.cz/full/148/4/mb148_4_1.pdf |
| work_keys_str_mv | AT jonathandavidfarley doestheendomorphismposetppdeterminewhetherafiniteposetpisconnectedanissueduffusraisedin1978 |