Here's an example of an algebraic approach to bitwise operations. I'm intentionally choosing something that is obvious from the programmer's point of view to see what it corresponds to algebraically.
We know that bitwise rotation of an n-bit word x can be done with left/shift shifts:
(x << 1) | (x >> (n-1)) = (x << 1) ^ (x >> (n-1)).
Algebraically, this means that the rotation operator C satisfies C = L + R^(n-1). Since C is a rotation it must satisfy C^n = 1, i.e. if we rotate n times we should end up where we started. This corresponds to the identity (L + R^(n-1))^n = 1.