Timeline for Ring with three binary operations
Current License: CC BY-SA 3.0
17 events
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| Jul 1, 2014 at 0:32 | comment | added | goblin GONE | "No, it's a commitment to talking about mathematical objects instead of presentations of mathematical objects." I have often wondered what mathematics would look like if this commitment was more widespread. | |
| Feb 6, 2013 at 17:33 | comment | added | Qfwfq | @NoahStein: appearently, one cannot have too much "distributivity": see my answer about Eckmann-Hilton theorem. | |
| Feb 6, 2013 at 16:42 | comment | added | Noah Stein | I think perhaps the sense in which this answer does not address the original question is that it shows ways of making lots of $n$-ary operations, but not ones which distribute over each other, which was perhaps the crux of the original question. | |
| Feb 6, 2013 at 5:36 | comment | added | Jan Weidner | I think this answer contains an interesting perspective, +1. | |
| Feb 6, 2013 at 4:19 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Feb 6, 2013 at 4:19 | comment | added | Qiaochu Yuan | @Martin: is it really that hard to see the two examples in the second paragraph? I will try to distinguish them more clearly. | |
| Feb 6, 2013 at 0:27 | comment | added | Martin Brandenburg | It seems to me that this "answer" doesn't really go into the question. | |
| Feb 5, 2013 at 23:16 | comment | added | Qfwfq | @Qiaochu: That a Lawvere theory is "generated in degree two" (in the sense of: "all the operations are determined by the operation $T^2\to T$") is an intrinsic fact, not an artifice of the presentation. Or am I missing something? | |
| Feb 5, 2013 at 20:51 | comment | added | Mariano Suárez-Álvarez | Well, you started with «The claim that there are two binary operations on rings is misleading», which is rather difficult to misunderstand, and my point is that that is sort of backwards. All the other operations that show up when you view rings as a Lawvere theory are the result of forcing rings into a Lawvere theory —this may be useful at times (it is useful at times!) but it is just a (mostly harmless) side effect of adopting a specific the point of view. | |
| Feb 5, 2013 at 20:09 | comment | added | Qiaochu Yuan | @Mariano: I think you misunderstood the point I was trying to make in the first paragraph. The first paragraph is context for the examples in the second paragraph. It was not intended to be an argument that when working with rings it is always necessary to talk about all of the available operations. That is, of course, silly. | |
| Feb 5, 2013 at 20:04 | comment | added | Mariano Suárez-Álvarez | «If it was good enough for Burnside, it is plenty good enough for me» is not surely not an argument, but it does carry some weight :-) | |
| Feb 5, 2013 at 20:04 | comment | added | Qiaochu Yuan | @Mariano: the examples are in the second paragraph, not the first. | |
| Feb 5, 2013 at 20:03 | comment | added | Mariano Suárez-Álvarez | Can you imagine the classification of finite simple groups done using (even the notation required to handle) all derived operations in a group? Unless «talking about mathematical objects instead of presentations of mathematical objects» serves a purpose, it is just formalities. And, sure, in some situations, it does serve a purpose. —in finding significative examples of ring-with-three-binary-operations, not so much! | |
| Feb 5, 2013 at 19:58 | comment | added | Qiaochu Yuan | No, it's a commitment to talking about mathematical objects instead of presentations of mathematical objects. Lawvere theories exist independent of a choice of generators in the same way that groups do. Some Lawvere theories (e.g. the Lawvere theory of smooth algebras) are best described all at once rather than using a presentation in the same way that some groups are. | |
| Feb 5, 2013 at 19:38 | comment | added | Mariano Suárez-Álvarez | But most of those infinite operations in a ring are derived ones. Counting them is sort of a display of love for formalities... | |
| Feb 5, 2013 at 19:33 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Feb 5, 2013 at 19:25 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |