Yes. Monotonicity holds, and multiplication by 1 is unital. The operator is symmetric and associative.
Exponentiation (say, by \(2\)) is a strict monoidal monotone.
\(1 = 2^0\) and \(2^x * 2^y = 2^{x+y}\)
No: monotonicity does not hold (multiplying negative numbers gives a larger number).