Motivation: A facebook post had a bunch of these as an 'intelligence' test, so I thought I would think about what this operation is. But I haven't done this Math in years!
I have an operation $\oplus$ defined by $a\oplus b = (a-b)||(a+b)$ where $||$ is the concatenation operation.
e.g.
$$5\oplus1=(5-1)||(5+1)=4|| 6 = 46$$ $$9\oplus3=(9-3)||(9+3)=6|| 12 = 612$$
What is this? What does it mean? So I have $f:\Bbb N\times\Bbb N \to \Bbb N$ here. Can someone tell me what sort of morphism this is? Or explain what this means mathematically? It is a binary operation I can see, but I am not sure what to call the mapping.
Let's look for identity: $5\oplus x=x\oplus 5=5$, means that $5\oplus x=(5-x)||(5+x)=5$ and associativity I don't even know how to start.
It isn't even a Magma since it isn't defined for all pairs $a,b\in\Bbb N$. If $a\oplus b$ is defined then $b\oplus a$ is not defined.
If you redefine the operation so that $a\oplus b=((|a-b|)\|(a+b))$ it would be a magma but not associative: $1\oplus (2\oplus 3)=1\oplus15=1416$ but $(1\oplus 2)\oplus 3=13\oplus3=1016$.