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Is there a special name for a binary operation on the set $S$ that is also into $S$, that is unambiguous with other uses. I.e. if it's "operator on $S$", I've heard that in other places meaning the operator could map to another set $T$. I'm working on a math library in the D language.

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A binary operation on a set $S$ is, by definition, a map $S\times S\to S$. Is that what you are looking for? –  Arturo Magidin Mar 25 '12 at 3:17
    
endomap: $S\times S \to S$ –  user2468 Mar 25 '12 at 3:21
    
BTW the tag (computer-science) is not relevant. The context was already set by OP in his/her sentence: ".. the D language." –  user2468 Mar 25 '12 at 3:25
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@J.D.: I don't like that: "endomap" suggests a map form a set to itself. –  Arturo Magidin Mar 25 '12 at 3:27

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Are you looking for the term closed binary operation? (other reference)

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"Closed binary operation" is rather strange terminology. Instead, mathematicians usually say that a (sub)set is closed under a binary operation. –  Bill Dubuque Mar 25 '12 at 3:57
    
@BillDubuque I find it strange too, but it makes sense. At least, it does not makes any other sense. Shall we conclude there is no unanimous terminology? There is in French and German: internal. –  jmad Mar 25 '12 at 4:30

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