Explanation for the definition of monomials as products of products I'm attempting to learn abstract algebra, so I've been reading these notes by John Perry. Monomials are defined (p. 23) as
$$
\mathbb{M} = \{x^a : a \in \mathbb{N}\} \hspace{10mm} \text{or} \hspace{10mm} \mathbb{M}_n = \left\{\prod_{i=1}^m{\left( x_1^{a_1}x_2^{a_2} \dotsm x_n^{a_n} \right)} : m,a_1,a_2,\dotsc,a_n \in \mathbb{N}\ \right\}
$$
which makes sense. I understand the non-commutativity of matrix multiplication, but I don't understand this statement (p. 24):

So multiplication of monomials should not in general be considered commutative. This is, in fact, why we defined $\mathbb{M}_{n}$ as a product of products, rather than combining the factors into one product in the form $x_1^{a_1}x_2^{a_2} \dotsm x_n^{a_n}$.

I'm missing the connection between the non-commutativity of multiplication and the construction of the monomial definition.

In the commutative case, $x_1x_2x_1$ is equal to $x_1^2x_2$. However, in the non-commutative case, they need not be equal. Suppose now that we're in a situation where $x_1x_2x_1 \neq x_1^2x_2$. If the set of monomials had not been defined as products of products, then $x_1x_2x_1$ would not be considered a monomial as it is not of the form $x_1^{a_1}x_2^{a_2}\dots x_n^{a_n}$.
