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What are the exact meanings of the terms "operation", "function" and "map" (are they even exactly defined)? I have always believed that an "operation" was a map

$$S \times S \times [...] \times S \rightarrow S.$$

However, according to the always-so-reliable Wikipedia, an operation can be any map on the form

$$A_1 \times A_2 \times [...] \times A_3 \rightarrow S.$$

Fine, but Wolfram's MathWorld disagrees with Wikipedia and agrees with my original definition. Which is correct?

Also, above I've used the word "map" to describe operations. Is this correct use of the term? Could I have replaced "map" with "function" without changing any of the meaning, i.e., do "map" and "function" mean exactly the same thing, or is there a slight difference in formal meaning and/or customary meaning?

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This is just a matter of clashing nomenclature. In General Algebra, "operation" is reserved for the first type you describe; in other contexts, it is used almost as a synonym for "function". So, they are both correct depending on the context. Likewise, "map": in some contexts, it is essentially a synonym of "function", but in others it can mean something entirely different (e.g., "The Four Color Map Theorem" is not about functions...) –  Arturo Magidin Mar 9 '11 at 1:13
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As Arturo says, it is a matter of definition. There is no one rulebook people follow on this kind of thing (making it difficult to catalog mathematical terminology in encyclopedias; what you see on Wikipedia is often just one possible choice out of many). The best any writer can do is make his or her definitions explicit before using them. For what it's worth, in a lot of usages, "function" and "map" are exact synonyms; in others, "map" may carry more information (e.g. for some topologists "map" may mean "continuous function"). One can't tell from the words; one must be told by the writer. –  anon Mar 9 '11 at 8:39
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