# Free monoids in Milne's Group Theory

I'm reading J.S. Milne's notes on Group Theory (highly recommended) and am attempting to understand the following;

Let $X$ be a set of symbols $X = \left\lbrace a, b, c, \dots \right\rbrace$ (possibly infinite) and let $S_X$ be the free monoid on $X$ (binary operation given by concatenation of words). When we identify a symbol $a \in X$ with a word $a \in S_X$, $X$ becomes a subset of $S_X$ and generates it (i.e. no proper submonoid of $S_X$ contains $X$).

I'm not entirely sure what is meant here. When he says "identify $a\in X$ with $a \in S_X$" does he mean "when we treat $a$ as a member of $S_X$"? Also, what is meant by "$X$ becomes a subset of $S_X$"? I understand that $S_X = \langle X\rangle$ because $S_X$ is built from the symbols in $X$ but I'm not sure how to justify that no proper submonoid of $S_X$ contains $X$.

• Different data types: $a\in X$ is a member of a set, $a\in S_X$ is a word of a monoid. Jun 13, 2017 at 14:47
• Technically, one could regard a symbol $a$ in $X$ and the one-symbol word $a$ as different objects. Milne is saying no, consider them to be the same thing. It's the same thing you do when you're constructing $\mathbf{Z}$ for example. An integer is really an equivalence class (for a certain equivalence relation) of pairs $(a,b)$ where $a$ and $b$ are natural numbers. But once the construction is complete, you say "Let's identify the natural number $n$ with the equivalence class of $(n,0)$." Technically, these are different things, but you do it so that you can regard $N$ as a subset of $Z$. Jun 13, 2017 at 14:49
• @user49640 I see, so when I write $S_X = \langle X \rangle$ I'm considering $X$ as a set of one-symbol words that generate $S_X$? Jun 13, 2017 at 14:51
• Right.${{{{{}}}}}$ Jun 13, 2017 at 14:52

As the comments explain, Milne is identifying $a\in X$ with $(a) \in S_X$ where I'm writing $(a)$ for a one element list/sequence/word. $(abc)$ would be a 3 element list. In category theory, there is a more general notion of subobject (and thus "subset") than the usual set-theoretic definition which is useful here. The notion of subobject instantiated for the category of sets and functions leads to a subobject of a set $Z$ being an injective function $\iota : W\to Z$. More precisely, it's an equivalence class of injective functions where $\iota_1 : W_1 \to Z$ and $\iota_2 : W_2 \to Z$ are equivalent if $\iota_1 = \iota_2\circ\varphi$ for some bijection $\varphi$. In fact, we can define a preorder on subobjects by saying $\iota_1\leq\iota_2$ when $\varphi$ is just an injection. Of course, every subset $Z'\subseteq Z$ in the usual sense determines the subobject via the inclusion $Z' \hookrightarrow Z$, and every injective function determines a subset $\iota[W]\subseteq Z$ via the direct image.
The upshot of all this is we can restate what Milne is saying without identifying anything. We're saying the map $\iota : X \to S_X$ defined as $\iota(x)=(x)$ is a subobject of $S_X$, and given any injective monoid homomorphism $f : M \to S_X$, if it "contains" $X$, meaning $\iota\leq f$, then $f$ is a bijection. This means we have some injective function $x \mapsto m_x : X \to M$ such that $f(m_x) = (x)$ at which point any element e.g. $(abc)\in S_X$ can be reconstructed as $f(m_a m_b m_c)$.