Subtraction ($x - y$) is a binary operator that also has a useful unary version ($-x$), which can be seen as a simple shorthand for the binary version ($-x \Leftrightarrow 0-x$). A unary version of the addition operator ($x+y$) can also be defined ($+x\Leftrightarrow 0+x$), but it is much less useful (even useless?).
It is also possible and useful to provide a unary version of division, where $/x$ is defined as $1/x$. Of course, the same can be done for multiplication, where $\times x$ could be defined as $1\times x$, but this seems as useless as the unary addition operator.
In Boolean algebra, one could also define a unary version of the "exclusive or" operator ($\oplus$), where $\oplus x$ could be defined as $1\oplus x$. This "unary exclusive or" is simply negation, usually written $\neg x$.
In general, it seems that deriving a unary operator from a binary one (written $\otimes$ below) is useful when the following properties are true:
- $\forall x, c \otimes (c \otimes x) = x$, for some constant $c$,
- $\forall x, c\otimes x\ne x$ (to make the unary operator actually useful).
My questions are the following:
Do the above properties and values ($c$ in particular) have standard names in mathematics? For the first two cases (subtraction and division), I realize that the constant $c$ is a neutral element for the operation, but it is a neutral element on the right, while here it's used on the left. And for the "exclusive or" example, 1 is not a neutral element.
Are there any other "well-known" binary operators for which it would be useful to define unary variants?