Find the number of different magmas that have $A$ as its underlying set

Question

I have a problem involving algebraic structures. Any help I can get here would be amazing.

Problem: We have a set $A$, $\text{card} A = n$, $n \in \Bbb N$. Find the number of different magmas that have $A$ as the underlying set.

(I hope I translated this correctly. Feel free to point out if something is wrong or not clear.)

My idea:

Following the definition, any operation I make needs to have the results within that set. So, for example, I imagined a table where we have some operation * and numbers $0$ and $1$ from set $\{0,1\}$: $$ \begin{array}{c|lcr} * & \text{0} & \text{1} & \\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \ \end{array} $$

That would be one operation with those two numbers. It depends how I define my operation, thus I can make more. I could make:

$ \begin{array}{c|lcr} *_1 & \text{0} & \text{1} & \\ \hline 0 & 0 & 0 \\ 1 & 0 & 0 \ \end{array} $ or $ \begin{array}{c|lcr} *_2 & \text{0} & \text{1} & \\ \hline 0 & 1 & 0 \\ 1 & 0 & 1 \ \end{array} $ or $ \begin{array}{c|lcr} *_3 & \text{0} & \text{1} & \\ \hline 0 & 1 & 1 \\ 1 & 1 & 1 \ \end{array} $

So, if I take $n$ numbers and place into this table, I could make $n$ operations. Would results be $n^{2^n}$=$n^{2n}$ ?

Thank you.

Edit: added more explanation.

Answer 1

The operation $*\colon A \times A \to A$ maps each (ordered) pair $(a, b)$ of elements of $A$ to some element $a * b \in A$. There are no restrictions at all on the operation, so that each pair $(a, b)$ has $|A| = n$ "options" for its image (independent of what the image of any other pair might be). The operation $*$ (and hence the magma on $A$) is completely determined by this mapping, and therefore, the number of possible magmas on $A$ is \begin{equation*} |A|^{|A \times A|} = n^{n^2}. \end{equation*}

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Alternatively, we can divide the reasoning into two parts in a more obvious manner.

1. The number of mappings from a finite set $X$ to a finite set $Y$, i.e., the number of functions of of the form $f \colon X \to Y$ is $|Y|^{|X|}$ (where $|X|$ and $|Y|$ denote the number of elements of $X$ and $Y$ respectively). Why? Each element $x \in X$ has $|Y|$ possible elements of $Y$ out of which one can be its image $f(x)$. This is independent of the what the image of any other element of $X$ is. To define the function $f$, it is necessary and sufficient to "assign" to each $x \in X$, and image $f(x) \in Y$, and this can be done in \begin{equation*} \underbrace{|Y| \times |Y| \times \cdots \times |Y|}_{|X|\ \text{times}} = |Y|^{|X|} \end{equation*}
2. Given any set finite $A$ of cardinality $n$, a magma on $A$ is fully determined by defining an binary operation on $A$, or in other words, an function $*\colon A \times A \to A$. Using the previous result, this can be done in $|A|^{|A \times A|} = n^{n^2}$ number of ways. This is the number of different (labelled) magmas on $A$.