[Math] Non-associative, non-commutative binary operation with a identity

Question

Can you give me few examples of binary operation that it is not associative, not commutative but has an identity element?

Answer 1

Here are a couple easy examples, I'll leave it to you to verify their properties.

The natural numbers where $n\ast m=n^m$.

The integers where $n\ast m=n-m$.

The real numbers without zero where $x \ast y = \frac{x}{y}$.