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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:

  1. $\forall x, c \otimes (c \otimes x) = x$, for some constant $c$,
  2. $\forall x, c\otimes x\ne x$ (to make the unary operator actually useful).

My questions are the following:

  1. 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.

  2. Are there any other "well-known" binary operators for which it would be useful to define unary variants?

Michel Schinz
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    Considering how often unary minus causes confusion, I don't think that we want more. A couple of days ago, there was another discussion of what $-1^2$ means. – badjohn Dec 27 '23 at 09:14
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    This might be going a bit away from what you're thinking of, but I find it useful to consider, each fixed $r \in \mathbb R,$ the "unary operator" (i.e. function) $f_r:{\mathbb R} \rightarrow {\mathbb R}$ defined by $f_r(x) = x+r$ to understand why we can right-side add $r$ to each side of an equation in real numbers and preserve equality. Similarly why we can left-side multiply each side of an equation in a group, scalar-product each side of an equation of vectors in a vector space, etc. Recall the uniqueness property of functions, namely $x=y \implies f(x) = f(y).$ (continued) – Dave L. Renfro Dec 27 '23 at 09:33
  • In these examples the functions are obtained from $2$-variable functions (i.e. from binary operators) by fixing one of the variables. – Dave L. Renfro Dec 27 '23 at 09:37
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    Another point to consider is that the binary operations for which it makes sense are generally convenient rather than fundamental. Look at abstract structures in algebra e.g. groups and fields. It is common to define addition and multiplication but not subtraction or division. If they are used then they are just a shorthand for addition or multiplication with the inverse. – badjohn Dec 27 '23 at 10:39

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