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Consider the following examples from which some definitions are derived:

Let us take an element from the set R of real numbers (say, the number 8) and another from the set L of lengths (say, 4m). Multiplication of both elements (8 x 4m) will render as result an element from the set L of lengths, namely, 32m. This constitutes an external binary operation of the first type with regard to the set of real numbers and the set of lengths. However, multiplication of two lengths (e.g. 4m x 6 m) no longer yields a length, but a surface (24m2). In such cases one speaks of an external operation of the second type with regard to the set of lengths.

I am translating the German terms äussere Verknüpfung der ersten Art (or der zweiten Art, depending).However, I have found he expression external binary operation somewhere. It did seem to refer to the second type. Could anyone confirm that? Would the first type be called unary operation?

I also found this comment: an external binary operation is a binary function from K × S to S. This differs from a binary operation in the strict sense in that K need not be S; its elements come from outside.

Does it hold for the other type? If not, where is the difference to so-called internal operation?

Thnaks in advance.

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No, the terms unary and binary only refer to the number of arguments taken by an operation. Usually, an operation on a (non-empty) set $A$ is a function $$ A^k \to A $$ for some $k \geq 0$. An operation is said to be nullary, unary, or binary if $k$ is $0,1$, or $2$, respectively, and $k$-ary otherwise.

The term external operation is rarely used, although the instance external product is somewhat common. It usually denotes a function $$ B \times A \to A $$ for two non-empty sets $A,B$. I've only seen this term used in special contexts, though, like for the product by scalars in a vector space.

On the other hand, functions $$ A \times B \to C $$ are rarely called operations. As far as I know, this usually happens when $A,B$, and $C$ are part of a graded ring and the function defines an operation on it, like in the case of the cup product. It is worth noting, though, that in many interesting cases these functions are operators.

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  • $\begingroup$ So would you recommend me to substitute the term external operation (which is what the English wikipedia provides for "äussereVerknüpfung) for external product then? $\endgroup$ – Javier Arias Jul 3 '15 at 20:17
  • $\begingroup$ Probably yes, but that depends on the context. Could you tell me more precisely what you are translating? If it is about products of length (or "powers of length") measurements, then it would probably be better to just call it product, dropping the "external" altogether. $\endgroup$ – A.P. Jul 3 '15 at 20:44
  • $\begingroup$ This is pretty much the context: de.wikipedia.org/wiki/Verkn%C3%BCpfung_(Mathematik) $\endgroup$ – Javier Arias Jul 4 '15 at 9:30
  • $\begingroup$ That provides the German terminology. Some of the paragraphs I quote i take from here, though (see the section External Binary Operations): en.wikipedia.org/wiki/Binary_operation $\endgroup$ – Javier Arias Jul 4 '15 at 9:39
  • $\begingroup$ Then there may be some discrepancy between German and English terminology. You could try and have a look to some German book that uses the term äussere Verknüpfung and look for an English translation. Personally, I've never seen the term external product (or operation) used for anything other than binary functions. In either case, note that your "external operation of second type" is not an operation at all, just like the dot product mentioned in your second link. $\endgroup$ – A.P. Jul 4 '15 at 10:19

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