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text: '[todo: make Free Group a parent of this]\n\n[todo: make the summary a bit better]\n[summary: A "word" over a set $X$ is a finite ordered list of elements from $X$ and $X^{-1}$ (where $X^{-1}$ is the set of formal inverses of the elements of $X$), as if we were treating the elements of $X$ and $X^{-1}$ as letters of an alphabet. A "freely reduced" word over $X$ is one which doesn't contain any consecutive cancelling letters such as $x x^{-1}$.]\n\nGiven a [3jz set] $X$, we can make a new set $X^{-1}$ consisting of "formal inverses" of elements of $X$.\nThat is, we create a set of new symbols, one for each element of $X$, which we denote $x^{-1}$; so $$X^{-1} = \\{ x^{-1} \\mid x \\in X \\}$$\n\nBy this stage, we have not given any kind of meaning to these new symbols.\nThough we have named them suggestively as $x^{-1}$ and called them "inverses", they are at this point just objects.\n\nNow, we apply meaning to them, giving them the flavour of group inverses, by taking the [set_union union] $X \\cup X^{-1}$ and making finite "words" out of this combined "alphabet".\n\nA finite word over $X \\cup X^{-1}$ consists of a list of symbols from $X \\cup X^{-1}$.\nFor example, if $X = \\{ 1, 2 \\}$ %%note:Though in general $X$ need not be a set of numbers.%%, then some words are:\n\n- The empty word, which we commonly denote $\\varepsilon$\n- $(1)$\n- $(2)$\n- $(2^{-1})$\n- $(1, 2^{-1}, 2, 1, 1, 1, 2^{-1}, 1^{-1}, 1^{-1})$\n\nFor brevity, we usually write a word by just concatenating the "letters" from which it is made:\n\n- The empty word, which we commonly denote $\\varepsilon$\n- $1$\n- $2$\n- $2^{-1}$\n- $1 2^{-1} 2 1 1 1 2^{-1} 1^{-1} 1^{-1}$\n\nFor even more brevity, we can group together successive instances of the same letter.\nThis means we could also write the last word as $1 2^{-1} 2 1^3 2^{-1} 1^{-2}$.\n\nNow we come to the definition of a **freely reduced** word: it is a word which has no subsequence $r r^{-1}$ or $r^{-1} r$ for any $r \\in X$.\n\n# Example\n\nIf $X = \\{ a, b, c \\}$, then we might write $X^{-1}$ as $\\{ a^{-1}, b^{-1}, c^{-1} \\}$ (or, indeed, as $\\{ x, y, z \\}$, because there's no meaning inherent in the $a^{-1}$ symbol so we might as well write it as $x$).\n\nThen $X \\cup X^{-1} = \\{ a,b,c, a^{-1}, b^{-1}, c^{-1} \\}$, and some examples of words over $X \\cup X^{-1}$ are:\n\n- The empty word, which we commonly denote $\\varepsilon$\n- $a$\n- $aaaa$\n- $b$\n- $b^{-1}$\n- $ab$\n- $ab^{-1}cbb^{-1}c^{-1}$\n- $aa^{-1}aa^{-1}$\n\nOf these, all except the last two are freely reduced.\nHowever, $ab^{-1}cbb^{-1}c^{-1}$ contains the substring $bb^{-1}$, so it is not freely reduced; and $aa^{-1}aa^{-1}$ is not freely reduced (there are several ways to see this: it contains $aa^{-1}$ twice and $a^{-1} a$ once).\n\n%%hidden(Alternative, more opaque, treatment which might help with one aspect):\nThis chunk is designed to get you familiar with the idea that the symbols $a^{-1}$, $b^{-1}$ and so on in $X^{-1}$ don't have any inherent meaning.\n\nIf we had (rather perversely) gone with $\\{ x, y, z \\}$ as the corresponding "inverses" to $\\{ a, b, c \\}$ (in that order), rather than $\\{ a^{-1}, b^{-1}, c^{-1} \\}$ as our "inverses" %%note:Which you should never do. It just makes things harder to read.%%, then the above words would look like:\n\n- The empty word, which we commonly denote $\\varepsilon$\n- $a$\n- $aaaa$, which we might also write as $a^4$\n- $b$\n- $y$\n- $ab$\n- $aycbyz$\n- $axax$\n\nFor the same reasons, all but the last two would be freely reduced.\nHowever, $aycbyz$ contains the substring $by$ so it is not freely reduced; and $axax$ is not freely reduced (there are several ways to see this: it contains $ax$ twice and $xa$ once).\n%%\n\n# Why are we interested in this?\n\nWe can use the freely reduced words to construct the [-5kg] on a given set $X$; this group has as its elements the freely reduced words over $X \\cup X^{-1}$, and as its group operation "concatenation followed by free reduction" (that is, removal of pairs $r r^{-1}$ and $r^{-1} r$). %%note:We make this construction properly rigorous, and check that it is indeed a group, on the [5kg] page.%%\nThe notion of "freely reduced" basically tells us that $r r^{-1}$ is the identity for every letter $r \\in X$, as is $r^{-1} r$; this cancellation of inverses is a property we very much want out of a group.\n\nThe free group is (in a certain well-defined sense from [-4c7]%%note:See [free_group_functor_is_left_adjoint_to_forgetful] for the rather advanced reason why.%%) the purest way of making a group containing the elements $X$, but to make it, we need to throw in inverses for every element of $X$, and then make sure the inverses play nicely with the original elements (which we do by free reduction).\nThat is why we need "freely-reducedness".',
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