Main Question or Discussion Point
According to both Wikipedia and Wolfram MathWorld, a binary operation must be closed. Wiki does leave room for a so-called external binary operation, i.e. a function from (K X S) to S, but not from (S X S) to K. (This would make the operators in physics actually "external operators", no? Do we edit all the physics books?) Wikipedia, for example, says in its site on binary operations that the inner product is hence not a binary operation. Yet elsewhere (on the site "inner product") Wiki defines the inner product as an "algebraic operation...". If the latter article is only fudging because the author didn't find the proper word, what word is appropriate for an inner product? My guess would be a map (since a map, being a function, can have domain ≠ codomain). Or does one just leave it as "relation"?