Talk:Algebraic operation
Add topicLatest comment: 10 months ago by D.Lazard in topic Arity
 | This article is rated C-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
| |||||||||||
Inconsistent definition of "algebraic operation"
[edit]The lead section of the dot product article describes the dot product as an algebraic operation, but the algebraic operation article limits the definition of algebraic operation to addition, subtraction, multiplication, and taking roots. Is there a factual error in either of these two articles? Jarble (talk) 02:42, 6 January 2015 (UTC)
- The article about completing the square appears to contain a similar contradiction. Jarble (talk) 02:54, 6 January 2015 (UTC)
- I'm going to go out on a limb here since I can't reference everything I'm going to say. The operations listed in this article have been called the "Fundamental Algebraic Operations" in at least one old source that I have and I am sure that I have seen the expression "basic algebraic operations" used for the same collection. This implies that there exist algebraic operations which are not basic (or perhaps we should say not atomic). Certainly a candidate for this more general type of algebraic operation would be a finite algorithm each of whose steps was an application of a basic algebraic operation. Again borrowing from chemistry, we could call such an algorithm a "compound algebraic operation". In this sense, both finding a dot product, completing the square, and a host of other computations (calculating a factorial, factoring a polynomial by grouping terms appropriately, come to mind) are algebraic operations. I don't think that I have ever seen this formally defined, but I am pretty sure that this is what most authors, who use this term loosely, would have in mind. A more modern approach would be to say that an algebraic operation is the result of applying an (algebraic) operator in some appropriate domain where the descriptor indicates that there exists some representation of the operator which only involves the basic algebraic operations. This is a little too fancy for most of the uses of this expression. Historically I think that the expression arose when a distinction was needed in taking operations on (real) numbers and expressing their corresponding action in the realm of algebra. This is a very limited use and I can see that there would be pressure to expand the envelope somewhat. I am not making a suggestion as to what to do with the articles in question, but I don't think that what is wrong with them can be called a factual error. Bill Cherowitzo (talk) 05:07, 6 January 2015 (UTC)
- I partly agree with Wcherowi. However, this article is about what is should be called (basic or fundamental) arithmetic operations. Thus, it should be renamed and/or merged into arithmetic. IMO, an algebraic operation is an operation of an algebraic structure, such are the arithmetic operations, and also the dot product, the product in a group, and so on. D.Lazard (talk) 16:16, 24 November 2015 (UTC)
- We can mention composition of permutations in permutation groups and geometric transformations as examples of algebraic operations. 2A02:2F07:DE10:B400:FD89:403B:3307:955E (talk) 17:13, 12 July 2025 (UTC)
- I partly agree with Wcherowi. However, this article is about what is should be called (basic or fundamental) arithmetic operations. Thus, it should be renamed and/or merged into arithmetic. IMO, an algebraic operation is an operation of an algebraic structure, such are the arithmetic operations, and also the dot product, the product in a group, and so on. D.Lazard (talk) 16:16, 24 November 2015 (UTC)
- I'm going to go out on a limb here since I can't reference everything I'm going to say. The operations listed in this article have been called the "Fundamental Algebraic Operations" in at least one old source that I have and I am sure that I have seen the expression "basic algebraic operations" used for the same collection. This implies that there exist algebraic operations which are not basic (or perhaps we should say not atomic). Certainly a candidate for this more general type of algebraic operation would be a finite algorithm each of whose steps was an application of a basic algebraic operation. Again borrowing from chemistry, we could call such an algorithm a "compound algebraic operation". In this sense, both finding a dot product, completing the square, and a host of other computations (calculating a factorial, factoring a polynomial by grouping terms appropriately, come to mind) are algebraic operations. I don't think that I have ever seen this formally defined, but I am pretty sure that this is what most authors, who use this term loosely, would have in mind. A more modern approach would be to say that an algebraic operation is the result of applying an (algebraic) operator in some appropriate domain where the descriptor indicates that there exists some representation of the operator which only involves the basic algebraic operations. This is a little too fancy for most of the uses of this expression. Historically I think that the expression arose when a distinction was needed in taking operations on (real) numbers and expressing their corresponding action in the realm of algebra. This is a very limited use and I can see that there would be pressure to expand the envelope somewhat. I am not making a suggestion as to what to do with the articles in question, but I don't think that what is wrong with them can be called a factual error. Bill Cherowitzo (talk) 05:07, 6 January 2015 (UTC)
Arity
[edit]Shouldn't arity be mentioned, since Cartesian power product is mentioned in the formal presentation of an operation? Cartesian power exponent indicates the arity of the operation, isn't it? 86.120.238.97 (talk) 14:28, 9 August 2025 (UTC)
- Using n-ary operation without defining what is n and using "n" instead of "" go agaisnt the manual of style. Linking Arity in such an elementary article without recalling what it is, goes also against Wikipedia standards. I would not oppose to define arity after the sentence linked to Cartesian product, or to rewrite this sentence for clarifying the relation between the exponent of the Cartesian product and arity. Note that the concept of arity is much more elementary than the concept of a Cartesian product, and WP:TECHNICAL would impose to talk of arity before of Cartesian products. D.Lazard (talk) 16:01, 9 August 2025 (UTC)
- We can phrase as:, the exponent of the Cartesian power/product equals the arity (number of terms/operands) of the operation. 86.120.238.97 (talk) 12:56, 31 August 2025 (UTC)
- As said above, "Cartesian poduct" is too technical for this lead. Nevertheless, I agree that arity deserves to be mentioned in the lead. I'll try to find a better formulation. D.Lazard (talk) 13:22, 31 August 2025 (UTC)
Done. D.Lazard (talk) 13:52, 31 August 2025 (UTC)
- As said above, "Cartesian poduct" is too technical for this lead. Nevertheless, I agree that arity deserves to be mentioned in the lead. I'll try to find a better formulation. D.Lazard (talk) 13:22, 31 August 2025 (UTC)
- We can phrase as:, the exponent of the Cartesian power/product equals the arity (number of terms/operands) of the operation. 86.120.238.97 (talk) 12:56, 31 August 2025 (UTC)
