Meaning of free commutative monoid and morphism

Question

Currently I am reading following paper on link: https://arxiv.org/pdf/math/0509164v1.pdf On page 2 of this paper, authors have defined $[X]$ as the free commutative monoid generated by $n$ variables $X=\{x_1,\cdots,x_n\}$. What is the meaning of this? In next line authors have defined a morphism from $[X] \to F_2^n$ , what is the meaning of the morphism. What kind of elements are there in $[X]$? i.e. how the elements in this set will look like? Is it an algebraic structure w.r.t multiplication?

Answer 1

$X$ is probably just the set of all so-called "formal sums" $$ a_1 x_1 + \ldots + a_n x_n $$ where the $a_i$ are nonnegative integers. The "sum" operation is defined by adding coefficients, and the zero element is the sum with all coefficients being zero.

If "formal sum" seems like nonsense to you, you could instead say that $X$ is the set of all functions from the set $\{1, \dots n\} \to \Bbb Z$ whose images consist of only nonnegative numbers. A typical function $f$ in this latter set corresponds to the formal sum in the earlier description via $a_i = f(i)$. Addition in this case is the usual addition of functions; the identity element is the constantly zero function.

As for the morphism: the obvious one would be

$$ (a_1, \ldots, a_n) \mapsto (a_1 \bmod 2, \ldots a_n \bmod 2). $$