Timeline for Why is every variety of bands determined by a single identity?
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
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| Apr 17, 2019 at 3:55 | review | Close votes | |||
| Apr 17, 2019 at 11:52 | |||||
| Jul 5, 2015 at 20:03 | answer | added | Keith Kearnes | timeline score: 10 | |
| Jun 7, 2015 at 23:41 | comment | added | John Baez | @PedroSánchezTerraf - thanks! What I really secretly wanted was a deeper understanding of universal algebra. I understand the general abstract nonsense aspects pretty well, but this result about bands is a nice example of a hard result that's not just general abstract nonsense. | |
| Jun 7, 2015 at 13:51 | comment | added | Pedro Sánchez Terraf | There is a (rather old) survey on universal algebra by Walter Taylor, available at math.uh.edu/~hjm/hjmathsurvey_1979.pdf, that discusses (among many other things) finitely and one-based equational theories. I don't think it will be of great help in this particular issue, but it might convey a bit of an eagle's eye view. | |
| Jun 6, 2015 at 20:56 | history | edited | John Baez | CC BY-SA 3.0 |
added 2 characters in body
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| Jun 6, 2015 at 20:43 | comment | added | John Baez | @Steinberg - Thanks, that reference is helpful though itself scary. It starts: "The description of the lattice of all varieties of bands by Birjukov, Fennemore and Gerhard represents a significant achievement in the study of semigroup varieties. Even though their papers have often been quoted, the concrete material in them was rarely referred to in view of the proofs which are either condensed or are too long. The list of representative identities set up by Fennemore was effectively used by Adair and Sukhanov. In brief, their results have been admired from a distance." | |
| Jun 6, 2015 at 1:39 | comment | added | Benjamin Steinberg | You might look at Gerhard, J. A.; Petrich, Mario Varieties of bands revisited. Proc. London Math. Soc. (3) 58 (1989), no. 2, 323–350 which claims to give a more conceptual approach to the lattice of band varieties. | |
| Jun 6, 2015 at 1:01 | comment | added | Benjamin Steinberg | If you go to quasivarieties the situation is different. The result of adjoining an identity to a 2-element left zero semigroup has no finite basis of quasi-identities. | |
| Jun 6, 2015 at 0:54 | comment | added | Benjamin Steinberg | @EWHLee may know. | |
| Jun 6, 2015 at 0:10 | comment | added | Benjamin Steinberg | Bands are a bit miraculous. I never understood this myself. | |
| Jun 6, 2015 at 0:09 | comment | added | Benjamin Steinberg | The variety generated by any finite group is defined by a single identity. This is a deep theorem of Powell - Oates. | |
| Jun 5, 2015 at 23:05 | answer | added | The Masked Avenger | timeline score: 2 | |
| Jun 5, 2015 at 22:18 | history | edited | John Baez | CC BY-SA 3.0 |
fixed error regarding number of identities
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| Jun 5, 2015 at 22:17 | comment | added | John Baez | I'm sorry, I forgot the associative law! I'll stick that in too. | |
| Jun 5, 2015 at 22:06 | comment | added | The Masked Avenger | From your perspective, that may be. From my perspective, I would include the associative law also, since I often look at other structures that have the signature < 2 > (one binary operation) . No examples come to mind that look as nice as bands. However, your question is about representing equational theories succinctly when they lie between the trivial theory and some (not necessarily finitely equationally based) nontrivial equational theory. I don't know the literature, but you may find Austin identities of (remotely) related interest. | |
| Jun 5, 2015 at 21:36 | comment | added | John Baez | I've clarified my statement a bit; I hope that helps. Every variety of bands is determined by one identity in addition to the identity $a a = a$ that all bands must obey. | |
| Jun 5, 2015 at 21:34 | history | edited | John Baez | CC BY-SA 3.0 |
clarification
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| Jun 5, 2015 at 21:32 | comment | added | The Masked Avenger | Do you mean exactly one identity? Or exactly one more than is needed to distinguish it as a subvariety? In the latter case, I'm confident there are more examples in other signatures. | |
| Jun 5, 2015 at 21:25 | history | edited | John Baez | CC BY-SA 3.0 |
added image
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| Jun 5, 2015 at 21:20 | history | asked | John Baez | CC BY-SA 3.0 |