Associative functions of real numbers with 1 and 0

Question

What are the binary functions $F$ of the real numbers, possibly taking an open subset or including infinity, that have an identity element and a zero element and are associative? I know that $F:[-\infty,\infty)^2\to\mathbb R, F(x,y)=x+y+m$ works with identity $-m$ and idempotent $-\infty$, and similarly $F:{\mathbb R}^2\to\mathbb R , F(x,y)=mxy$, but are there any others?