$A = \{0, 1, 2, 3, \dots\}$ is the set of non-negative integers. Let F be the set of functions from $A$ to itself. For any two functions, $f_1, f_2 \in \text{F}$, we define $$(f_1 \odot f_2)(n) = f_1(n) + f_2(n)$$ for every number $n$ in $A$. Which of the following is/are CORRECT about the mathematical structure $(\text{F}, \odot)$?

Let $A = \{0, 1, 2, 3, \dots\}$ and $F = \{f \mid f : A \to A\}$. The binary operation $\odot$ on $F$ is defined pointwise as $(f_1 \odot f_2)(n) = f_1(n) + f_2(n)$ for all $n \in A$.

Let us evaluate the algebraic properties of $(F, \odot)$:

1. Closure: For any $f_1, f_2 \in F$ and $n \in A$, both $f_1(n) \in A$ and $f_2(n) \in A$. Since the sum of two non-negative integers is a non-negative integer, $(f_1 \odot f_2)(n) = f_1(n) + f_2(n) \in A$. Hence, $f_1 \odot f_2 \in F$.

2. Associativity: For any $f_1, f_2, f_3 \in F$ and $n \in A$:

   $$((f_1 \odot f_2) \odot f_3)(n) = (f_1(n) + f_2(n)) + f_3(n) = f_1(n) + (f_2(n) + f_3(n)) = (f_1 \odot (f_2 \odot f_3))(n)$$

   Associativity holds because addition of integers is associative.

3. Commutativity: For any $f_1, f_2 \in F$ and $n \in A$:

   $$(f_1 \odot f_2)(n) = f_1(n) + f_2(n) = f_2(n) + f_1(n) = (f_2 \odot f_1)(n)$$

   Commutativity holds because integer addition is commutative.

4. Identity Element: Define $e : A \to A$ by $e(n) = 0$ for all $n \in A$. Since $0 \in A$, $e \in F$. For any $f \in F$:

   $$(f \odot e)(n) = f(n) + e(n) = f(n) + 0 = f(n)$$

   Thus, $e$ serves as the identity element under $\odot$.

5. Inverses: For an inverse $g \in F$ of a function $f \in F$, we require $(f \odot g)(n) = e(n) = 0$, meaning $f(n) + g(n) = 0$ for all $n \in A$. Since $f(n), g(n) \ge 0$, this requires $g(n) = -f(n)$. If $f(n) > 0$ for any $n$, then $g(n) < 0 \notin A$, so $g \notin F$. Thus, inverses do not exist for all elements in $F$.

Since $(F, \odot)$ is associative, commutative, has an identity element, but lacks inverses for all elements, $(F, \odot)$ is an Abelian monoid.

Therefore, option B is correct.