Consider "False", "maybe", "True" group table. (1) "True" is a unitary element upon AND. (always returns the other element) (2) "False" is a unitary element upon OR.
I feel, I am missing an operation that makes "maybe" a unitary element.
That is (@) "maybe?X=X?maybe = X".
Is there such operation over this group?
There are many approaches to fuzzy logic. What you are looking for can be found here. The truth value of a statement $p$ is given by $$t\in[0,1]$$ which means: If $t=0$, the statement is absolutely false
If $t=1$ the statement is absolutely true
and $t\in (0,1)$, the higher $t$ is, the more likely the statement is true which is your maybe.