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 $x_1^{a_1}x_2^{a_2}\dots x_n^{a_n}$.