Timeline for Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?
Current License: CC BY-SA 2.5
19 events
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| Jan 3 at 15:23 | comment | added | Joel David Hamkins | @DanielDonnelly By operation I simply mean a binary function on the integers $(m,n)\mapsto m\star n$, which takes two integer inputs $m$ and $n$ and has value $m\star n$. The question is not about run-time or computational complexity, but rather about the whether there is such an operation $\star$ for which $m+n$ and $m\cdot n$ can both be expressed entirely as terms using $\star$ only. | |
| Jan 3 at 2:11 | comment | added | Daniel Donnelly | You haven't correctly identified what is allowed as an "operation". For example, you implicitly assume we all consider $\text{id}(x)$ to be a non-operation. But on the other hand $-x$ which is usually considered unary. Otoh, if you have identified all $-x$ in some finite group say, then that operation is similarly only $O(1)$ in run time as $\text{id}(x)$ which we'll call memory look up. So I will only answer if you lay out some rules, otherwise I'm voting to close as clearly too broad. | |
| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Sep 23, 2014 at 3:42 | answer | added | user44143 | timeline score: 5 | |
| May 13, 2011 at 17:37 | vote | accept | Joel David Hamkins | ||
| May 13, 2011 at 16:14 | answer | added | Someone | timeline score: 4 | |
| May 12, 2011 at 17:02 | answer | added | Goldstern | timeline score: 33 | |
| Mar 5, 2011 at 20:25 | answer | added | Gerhard Paseman | timeline score: 6 | |
| Mar 5, 2011 at 20:13 | answer | added | Terry Tao | timeline score: 18 | |
| Mar 5, 2011 at 19:55 | answer | added | Andrej Bauer | timeline score: 16 | |
| Mar 5, 2011 at 19:14 | answer | added | Terry Tao | timeline score: 39 | |
| Mar 5, 2011 at 18:56 | comment | added | Joel David Hamkins | Asaf, indeed, one can easily define the operations by formulas of first order logic as in your exercise. In my question, however, I am seeking a binary operation having mere terms that give rise to the given functions. | |
| Mar 5, 2011 at 18:43 | comment | added | Gerhard Paseman | There is a notion of universal term. When I get to my copy of ALV, I will look it up. I remember it as an exercise as well as work done by George McNulty. Gerhard "Ask Me About Clone Theory" Paseman, 2011.03.05 | |
| Mar 5, 2011 at 18:20 | answer | added | Gerald Edgar | timeline score: 1 | |
| Mar 5, 2011 at 18:18 | comment | added | José Figueroa-O'Farrill | Although most probably unrelated to your question, something of this ilk happens in the context of Poisson algebras. In a Poisson algebra there are two operations: a commutative, associative multiplication and a Lie bracket, subject to a compatibility condition. It is possible to unify them into one product whose symmetric part is the commutative multiplication and the antisymmetric part is the Lie bracket. | |
| Mar 5, 2011 at 18:11 | answer | added | Guntram | timeline score: 0 | |
| Mar 5, 2011 at 16:46 | comment | added | Asaf Karagila♦ | In the introductory course I was TA'ing last semester we gave an exercise to define $\le$, $+$ and $\cdot$ from $a\mod b$ (when everything is zero when taken mod 0). | |
| Mar 5, 2011 at 16:09 | comment | added | M.G. | Very interesting question! (By some coincidence I was wondering about a similar thing myself too...) | |
| Mar 5, 2011 at 15:35 | history | asked | Joel David Hamkins | CC BY-SA 2.5 |