[Math] Can an element other than the neutral element be its own inverse

Question

Take the following operation $*$ on the set $\{a, b\}$:

• $a * b = a$
• $b * a = a$
• $a * a = b$
• $b * b = b$

$b$ is the neutral element. Can $a$ also be its own inverse, even though it's not the neutral element? Or does the inverse property require that only the neutral element may be its own inverse but all other elements must have another element be the inverse.

Answer 1

Yes, an element other than the identity can be its own inverse. A simple example is the numbers $0,1,2,3$ under addition modulo 4, where 0 is the identity, and 2 is its own inverse.