Talk:Binary operation

There's also binary function. Merge them? -- JanHidders

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I'm not sure they really mean the same thing. A binary operation is usually an algebraic operation, and is often denoted more like a*b than f(a,b). Probably the article ought to explain this. Also, if I had written the binary operation article from scratch I would have only allowed it to cover functions of the form f : S x S -> S, rather than the general f : S x T -> U. I didn't like to change the original too much, but perhaps it should be changed. In any case it would be a good idea to cross-link binary function and binary operation.
Zundark, 2001-08-08

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I agree, binary operations are S x S -> S. This article simply describes functions with two arguments. I think it should be changed, and the popular infix notation a*b for *(a,b) should be mentioned. --AxelBoldt

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Oh, you guys don't consider the vector scalar product (V * V -> R) or scaling of vectors (R * V -> V) or matrices ( R * M -> M ), etc to be binary operations? --Buz Cory

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Perhaps we should ask "what would Eric Weisstein" have done?" :-) But he doesn't seem to be sure either. There is

http://www.google.com/search?q=cache:LSEQ3bRAKkI:br.crashed.net/~akrowne/crc/math/b/b211.htm+binary+operator+eric+weisstein&hl=en

and there is

http://www.google.com/search?q=cache:FjdWL0BZpB4:www.math.pku.edu.cn/library/encyclopedia/contents/BinaryOperator.html+binary+operator+eric+weisstein&hl=en

which doesn't explicitly require the input domains to be the same. I know that in my own field (computer science) the term is used for any operator that needs two arguments. Perhaps it should be someting like this:

1. begin with S x S -> S definition
2. something about the notation
3. a remark that sometimes also S x T -> U is possible, with Buz' examples and ref. to binary function

-- JanHidders

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Maybe we should distinguish between a binary operation on a set (S x S->S) and a binary operation as such (S x T->U)? I don't know. --AxelBoldt

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I think you are onto something, Axel. Binary operation on a set requires closure for the result, and the elements chosen must also be from the set, so (S x S ->S) makes more sense. WMORRIS

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My textbook doesn't agree with the definition used on this article. I guess it is rather a convention or terminology problem than a real issue. It defines a binary operation as "f:AxA -> B". Where closure isn't required. The definition of a Group includes the requirement for closure of course, again, conflicting with the Group article. This is the convention in Israel, I guess.. -- Rotem Dan 14:20 13 Jul 2003 (UTC)

Well, from further research, it is actually quite unique to my university. Popular definition in most universities in Israel is f:AxA -> A --Rotem Dan 09:00 26 Jul 2003 (UTC)

At present, there seems to be a compromise between the 'Only operations on a set' and 'General binary functions' alternatives; there is an added section covering some of the 'other' binary operations. However, it does not solve the issue. First of all, I wonder if there hasn't been a confusion between 'scalar product', mentioned as an example supra, and 'scalar multiplication', which now is given as an example in the article. In classical terminology, 'scalar product' is a function $V\times V\rightarrow R$ (where V is a vector space over the scalars R, in this context mostly the real numbers); while 'multiplication with scalars' is a function $R\times V\rightarrow V$. Some (but far from all) modern textbooks instead use the terms 'dot product' and 'scalar multiplication', respectively. In my opinion, both functions are legitimate candidates for the term 'binary operation'.

Actually, the restriction to the two singly enumerated instances in the present article are not only in conflict with some literature, but with a number of Wikipedia items in the category 'Binary operation'; e.g., Commutative_operation and Outer_product.

Therefore, if no one protests, I think we should change the item, noting that the term is used in different senses in different contexts, sometimes very broadly, including in the Wikipedia notes. If any one does protest, I suggest that he or she briefs through all the items in the category, and lists those that should be omitted or rewritten, if we are to retain the present restricted definition. JoergenB 13:44, 27 August 2006 (UTC)

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I would like to see a discussion of the extension of a binary operator to finite sequences through repeated application. For example, the addition operator can be extended to the sum operation, the multiplication operator to the product operation, etc. In general, any binary operator (+) with a left identity can be extended to an operation on finite sequences whose value on the empty sequence is the left identity and whose value on a sequence {a[i]:0 <= i < k+1} of length k + 1 is Sk (+) a[k], where Sk is the value of the operation on the leading subsequence (prefix) {a[i]:0 <= i < k} of length k.

Moreover, what I would really like to see is the generic name for this new operation, which is what I was looking for when I came to this page. I've found the terms "bulk action", "iterated binary operation", and "prefix operation" through google, but haven't seen any clear evidence that any of these terms is in common usage. NoJoy 18:36, 28 October 2005 (UTC)

I don't think "iterated binary operation" would belong in a page about binary operations since it requires a unique left identitity and in the prototypical cases of sum and product notation requires associativity. However a link to such a page would be appropriate if someone who knows enough about it is willing to write it. TooMuchMath 01:50, 13 February 2006 (UTC)

OK, I bit the bullet and added a new page myself. It probably needs help. NoJoy 19:05, 8 June 2006 (UTC)

I think you're describing "folding". This is a common notion in functional programming languages, used for recursing (iterating) over data types such as sequences and trees. The only description I can find in Wikipedia is the article Catamorphism, which is the same concept disguised by category theory. However, there should be plenty of stuff on the web if you search for "fold" and "unfold". --Malcohol 10:40, 30 August 2006 (UTC)

Oh! I just found your article Iterated_binary_operation which does link to Fold (higher-order function).--Malcohol 10:44, 30 August 2006 (UTC)

--No mention of blob-- The symbol? —Preceding unsigned comment added by 88.106.125.175 (talk) 21:35, 26 February 2008 (UTC)

Closure

It is very tough to determine whether or not closure is necessary in a binary operation from this article. The first paragraph leads one to believe that closure is not required, but then the more precise definition that follows leads one to believe closure is required. Then the article flips back and describes situations where closure is not required. I think the real issue is that the term has been overloaded such that it means slightly different things in different contexts, but this should somehow be made more clear. Mickeyg13 (talk) 17:15, 9 May 2011 (UTC)

Commutivity

Is the line: "If the operation is commutative, ab = ba, then the value depends only on the multiset a,b,c." Meant to have (a,b),c, where a and b must be together. For it to not matter where c comes in terms of a and b, i.e between a and b, then wouldn't it also require associativity, where it doesn't matter if you do a and b first, or b and c? The next line states it depends only on the multiset a,b,c if it is both associative and commutative.— Preceding unsigned comment added by 149.171.197.62 (talk • contribs)

I fixed it, intended formatting changed the meaning.--Patrick (talk) 07:24, 8 August 2011 (UTC)

Mistake?

The article says "More precisely, a binary operation on a set S is a binary relation that maps elements of the Cartesian product S × S to S"

Isn't a binary operation supposed to be a function? This isn't even mentioned in the article. Instead it says it is a binary relation, wich is not a function so should be wrong.