Let $u_1,u_2 \in \mathbb{R}$. Let us define the binary operation $*$ as
$$u_1*u_2=u_1+u_2-\ln\left[ \exp^{u_1}+\exp^{u_2} \right].$$ It can be checked that this operation is commutative and associative which makes it a semi-group.
How can I find its kernel, sub-semi-groups and other substructures ("minimal blocks")?
As $$ e^{a*b}=\frac{e^ae^b}{e^a+e^b}$$ we obtain magma isomorphism $\langle\Bbb R,*\rangle\to \langle (0,\infty),\circ\rangle$, $x\mapsto e^x$, where $\circ$ is defined as $$ a\circ b = \frac{ab}{a+b}.$$ We can take this further by noting that $$ \frac1{a\circ b}=\frac1b+\frac1a.$$ Thus, we obtain an isomorphism of magmas $$\begin{align}\langle R,\circ\rangle&\to \langle (0,\infty),+\rangle\\x&\mapsto e^{-x}\end{align}$$ and thus can infer all we want to know about $\langle R,\circ\rangle$ from the better-known $\langle (0,\infty),+\rangle$. (In particular, we infer without direct computation that the magma $\langle R,\circ\rangle$ is in fact a abelian topological semigroup without a neutral element, with cancellation law, etc.