It I have a set $W$ that has a binary function $f$ defined on it maps 2-tuple combinations of $W$ onto another set $K$, i.e,
$ f: W \times W \rightarrow K$
Here f is non-surjective and non-injective. Let us also introduce another function $g$ that maps $W$ onto another set $V$, i.e,
$ g : W \rightarrow V$
Here let $g$ be both surjective and injective (aka bijective). Lastly introduce another function $y$ that maps 2-tuple combinations of $V$ onto $K$, i.e.
$ y : V \times V \rightarrow K$
With the property
$f(w_{1}, w_{2}) = y(g(w_{1}), g(w_{2}))$
Then what properties must exist on $W$, $K$, $V$, $g$, $f$ and $y$ so that the above statement holds?
Any guidance would be greatly appreciated as well as any good online tutorials, textbooks, Itunes U Units, etc that can help with this sort of stuff?
Thanks in Advance, David
If $W,K,V,f$, and $g$ are given, you can simply define $y:V\times V\to K$ to have the desired property by setting
$$y(v_1,v_2)=f\big(g^{-1}(v_1),g^{-1}(v_2)\big)$$
for each $\langle v_1,v_2\rangle\in V\times V$. This makes sense, since $g^{-1}$ is a bijection from $V$ to $W$, and it implies that
$$f(w_1,w_2)=f\big(g^{-1}(g(w_1)),g^{-1}(g(w_2))\big)=y\big(g(w_1),g(w_2)\big)$$
for each $\langle w_1,w_2\rangle\in W\times W$.