If $\mathsf{X}$ is a set, a binary operation on it is a map $*:\mathsf{X}\times\mathsf{X}\to\mathsf{X}$. Examples of these operations are addition and multiplication in a field.

The scalar multiplication in a vector space $\mathsf{V}$ over a field $\mathsf{F}$ is defined as a map $\cdot:\mathsf{F}\times\mathsf{V}\to\mathsf{V}$. This doesn't fit into the definition of a binary operation unless $\mathsf{V}=\mathsf{F}$. What is it called? Is there a term for it?

  • $\begingroup$ Aaah found it! It's called a left external binary operation on $\mathsf{V}$ over $\mathsf{F}$! $\endgroup$ May 27 at 14:20
  • 1
    $\begingroup$ The answer provided below is much more common. $\endgroup$
    – Randall
    May 27 at 14:31

1 Answer 1


"Action" or more generally a "left-action"


  • $F$ acts on $V$
  • An action of $F$ on $V$
  • A group action

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