Defining homomorphism in a function.

Question

The resource is present here, page 3. They define a function -

$F((w_{i}^{T} - w_{j}^{T}).w_{k}^{'}) = \dfrac{P_{ik}}{P_{jk}}$

where $P_{ik} = \dfrac{X_{ik}}{X_{k}}$, $X$ is a 2-D matrix, $w$ are row vectors of matrix $W$, and $w^{'}$ is row vector of matrix $W^{'}$.

Now they are stating that they need to interchange $w \leftrightarrow w^{'}$ and $X \leftrightarrow X^{T}$ to make the equation symmetric, and they use group homomorphism between $(R,+)$ and $(R_{>0}, x)$.

After which they state this as equation 4 -

$F((w_{i}^{T} - w_{j}^{T}).w_{k}^{'}) = \dfrac{F(w_{i}^{T}w_{k}^{'})}{F(w_{j}^{T}w_{k}^{'})}$

Questions -

1. Why are they specifically choosing addition and multiplication, and not other symbols? I can see addition but not multiplication.

2. How are they achieving equation 4 stated there? Since if they are using it over multiplication, then why aren't both the F's on the r.h.s should have been multiplied?

Answer 1

What they are asking is for $F$ to be a group homomorphism $(\Bbb R,+,0)\to (\Bbb R,\times,1) $. A consequence of this is that \begin{align}F((w_i^T-w_j^T)\cdot w_k') & =F(w_i^T\cdot w_k'-w_j^T\cdot w_k')=F(w_i^T\cdot w_k'+(-w_j^T\cdot w_k'))\\ & =F(w_i^T\cdot w_k')\times F(-w_j^T\cdot w_k') = F(w_i^T\cdot w_k')\times F(w_j^T\cdot w_k')^{-1}\\ & = \dfrac{F(w_{i}^{T}w_{k}^{'})}{F(w_{j}^{T}w_{k}^{'})},\end{align} since group homomorphisms preserve inverse, and $-w_j^T\cdot w_k'$ is the inverse of $w_j^T\cdot w_k'$ in $(\Bbb R,+,0)$. The multiplication is in some sort "hidden" in the division in R.H.S.