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Old 06-27-2019, 12:28 PM
Derleth is offline
Join Date: Apr 2000
Location: Missoula, Montana, USA
Posts: 21,182
Originally Posted by Topologist View Post
Here's a specific case where that makes a difference. I often work with systems that are anticommutative, meaning that there's a multiplication with yx = -xy. Looking at the case where y = x, that means that x2 = -x2. That's fine and always true mod 2, but mod an odd prime it means that we have to have x2 = 0. The latter puts a significant restriction on what anticommutative structures are possible mod an odd prime that isn't present working mod 2.
Interesting. I'm familiar with Clifford algebras, specifically geometric and spacetime algebra, which are simply the three-dimensional Clifford algebra and the four-dimensional Clifford algebra with the Minkowski metric, respectively. In those algebras, vector multiplication is anticommutative but x2 for x some vector of grade 1 or higher isn't 0, it's simply the dot product; if x is a basis vector, it's either 1 or -1, depending on the metric.

The only place I already knew about where x2 = 0 is in a form of nonstandard analysis where d2 = 0, such that differentiation can be done algebraically.
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