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Old 06-27-2019, 07:02 PM
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Originally Posted by Chronos View Post
Aren't the integers mod 2 a pretty trivial arithmetic to be doing anything with, though?
Not when you're using the field F2 with two elements as your base field and working with algebras over F2 or the like. Lots of very rich structures possible. Think of them as algebraic systems in which twice something happens to always be 0.

Originally Posted by Derleth View Post
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.

I deliberately left out a lot of detail. In algebraic topology we often work with rings graded on the integers (or more exotic things), and those rings are anticommutative in the sense that yx = (-1)deg x deg y xy. So you get a negative sign if both x and y lie in odd grading, but if either lies in even grading the product is commutative. This means, for example, that if you want to define something like a polynomial algebra Fp[x] with x in a nonzero grading, if p is odd and you want x to have grading 1, you can only get a truncated algebra, with x2 = 0. On the other hand, if p = 2, you can have x in grading 1 and get a full polynomial algebra.