Talk:Hall word

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There's something basic I'm not understanding about this article that makes it frankly illegible to me. Commutators figure prominently in the first section, but the setting is a free associative magma, and the definitions of commutator in a ring (${\displaystyle xy-yx}$) and in a group ${\displaystyle xyx^{-1}y^{-1}}$ don't seem to be available. This makes the argument difficult to meaningfully follow. vivacissamamente (talk) 01:04, 10 November 2020 (UTC)[reply]

The setting of a magma is exactly what makes all of this work. Review the article on magma (algebra). It just says that, for a set X containing letters a,b,c,... one can always write x.y for x and y being any of the letters a,b,c,... The magma has no further identities or relations: no commutativity, no associativity, no identity element, nothing. What this means is that whenever you see x.y (in the magma) you can ("mentally") replace x,y by [x,y] which is "the same thing". And if you feel like writing [x,y]=xy-yx or if you wish to write ${\displaystyle x.y=xyx^{-1}y^{-1}}$ that's up to you, you can do that, too. You are free to make these substitutions aka "rewrites". However, as a general rule, you want to do this last, after everything else, as otherwise you'll get carpet burns trying to keep track of the intermingled juxtapositions and group operations. I'll try to add a clarifying remark to the article. 67.198.37.16 (talk) 03:00, 30 November 2023 (UTC)[reply]

Well, OK, done. Perhaps three years too late, but as they say, better Nate than lever. 67.198.37.16 (talk) 03:39, 30 November 2023 (UTC)[reply]