Solution
= Solution
The substitution $F(a)=e^a$, with $F(-\infty)=0$, is a <bijection> onto $[0,\infty)$ and obeys
$$
F(a*b)=F(a)+F(b).
$$
Thus the <log-sum-exp function> transfers nonnegative addition to the given set. It is closed, associative and commutative, with <identity element> $-\infty$. But a finite $a$ has $e^a>0$, and adding a nonnegative number cannot give zero, so no finite $a$ has an <inverse element>. Therefore it is a <commutative monoid> but \b[not a group].