Talk:Kernel (algebra)

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Kernel (algebra) is currently a Mathematics and mathematicians good article nominee. Nominated by Gramix13 (talk) at 03:38, 23 May 2025 (UTC)

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Short description: Elements taken to zero by a homomorphism

Untitled

Considering they are directly related, no reason not to have them merged.

Null space and kernel are taught as different concepts in different contexts. --Bkwillwm 21:45, 5 December 2005 (UTC)[reply]

I would probably support a merger of all three of kernel (algebra), kernel (mathematics, and null space. -lethe ^{talk +} 09:47, 14 March 2006 (UTC)[reply]

I suggest the merge discussion be centralized in kernel (mathematics). — Arthur Rubin | (talk) 02:57, 17 March 2006 (UTC)[reply]

Convolution

Could someone explain how this is or is not the same as the kernel of a convolution? It seems to be different in that a convolution is linear yet in general only the zero vector maps to zero (I think). —Ben FrantzDale 15:47, 20 December 2006 (UTC)[reply]

picture

can the editor of this article give me suggestions on how to make this picture consistent with the terminology of this article, thanks--Cronholm144 09:27, 31 May 2007 (UTC)[reply]

What is a Mal'cev algebra?

The article speaks of groups as being Mal'cev algebras, but the linked article Malcev algebra describes something that groups definitely are not, so what is a Mal'cev algebra in the context of this article here? 95.88.237.20 (talk) 19:13, 13 February 2013 (UTC)[reply]

Surely additive groups are Mal'cev algebras when equipped withe the zero product. But Mal'cev algebra is too much a stub to decide what the editor did mean. I have tagged Mal'cev algebra as stub? D.Lazard (talk) 21:22, 13 February 2013 (UTC)[reply]

Aren't Mal'cev algebras supposed to be non-associative algebras, which in turn are supposed to be modules? If a module needs to be an abelian group, how can a non-abelian group be a Mal'cev algebra? 95.88.237.20 (talk) 12:26, 14 February 2013 (UTC)[reply]

I have changed "group" into "commutative group". I hope that ths solves the problem. D.Lazard (talk) 15:23, 14 February 2013 (UTC)[reply]

I think this was intended to refer to Mal'cev algebras in the sense of general algebra, which are indeed structures for which knowing the kernel in the simpler sense suffices to construct the whole kernel: specifically, all reflexive relations closed under the operations are automatically equivalence relations. But I don't know what the original editor meant about a neutral element. — Preceding unsigned comment added by 2605:E000:844F:6600:35E1:AACB:A137:99D5 (talk) 01:36, 8 December 2016 (UTC)[reply]

Apparently, Mal'cev algebra refers to a generalization of groups that has neutral element

e

, and some kind of subtraction such that, if

f

is a homomorphism, then

f (a) = f (b)

is equivalent with

f (a - b) = e

. I do not know which axioms this operation should have for having this property. I have tagged the section as "confusing", as it cannot be understood without a correct definition. D.Lazard (talk) 09:09, 8 December 2016 (UTC)[reply]

Kernel (Ring homomorphisms)

This section incorrectly states "It turns out that, although ker f is generally not a subring of R," but this is false (http://www.proofwiki.org/wiki/Kernel_is_Subring). I'll review the symbology carefully and update if it turns out that the article is in fact erroneous. 129.21.72.58 (talk) 01:10, 28 April 2013 (UTC)[reply]

You are wrong, the article correctly asserts that the kernel is a non-unital ring but is not a unital ring, unless if the target of the homomorphism is the zero ring. D.Lazard (talk) 08:37, 28 April 2013 (UTC)[reply]

More accessible initial explanation

Knucklehead here. I was reading about kernel methods and I wondered what a kernel was mathematically. So I came here, and the intro is incomprehensible to me. I'm wondering if those fond of this page might consider using plainer, less technical language to describe what it is initially. Examples of clearer explanations from the web are https://mathworld.wolfram.com/Kernel.html and https://www.quora.com/What-is-a-kernel-in-mathematics-Is-it-like-a-function . I disagree with the respondent on Quora, who said "To understand what "kernel" means in math, you need some background. It makes no sense to try and learn these things from Wikipedia. This is like trying to learn a language using only a dictionary." Nonsense: it makes sense to try to learn what the concept is on Wikipedia, and Wikipedia has far more utility than a dictionary. Ironically, he then proceeds to explain it fairly well like someone might do on Wikipedia, and better than this page currently does for me. Crucially, more accessible explanations will not use such technical terms as "homomorphism" or "injective". I comment here in hopes that more knuckleheads can appreciate mathematical concepts that are actually pretty accessible. 174.52.240.90 (talk) 23:00, 15 April 2020 (UTC)[reply]

I have rewritten the lead for making it less technical (and also less vague). In particular, I have added a definition for the unavoidable technical terms. D.Lazard (talk) 08:45, 16 April 2020 (UTC)[reply]

Pre-review

@Gramix13 since you are new to GAN, I think I can give some advice that, whenever one writes mathematics, do not use "we" and try to rephrase (see MOS:MATH#NOWE). Also, I can give a spot of citation needed. Feel free to improve by adding citations or revert for the problems related to WP:CALC. Dedhert.Jr (talk) 02:22, 24 May 2025 (UTC)[reply]

Hi Dedhert.Jr, and thanks for the advice you (and others) have given as pre-review. I will acknowledge that I am still new to editing Wikipedia, so I am open to feedback on ways to improve writing in articles that I edit.

Part of the reason I made an account was seeing an opportunity to vastly improve on this article as I initially saw it in Start-class condition, with Good Article as a goal. I'll admit that I haven't taken the time to review MOS:MATH (I did review the other MOS that were relevant to GAN to try my best at satisfying those), so I'll spend some time doing so to make sure it complies.

When I started working on this article, there were only two citations and they were really only used in the lead of the article for a single footnote (see [1]). If you're willing to check on places in the article that warrant citations, I would greatly appreciate the help.

Could you clarify where in the article there may be issues with WP:CALC so I can address them? The only apparent calculation I can think of would be the proof of a group homomorphism being injective if and only if its kernel is the singleton set with the identity, although since that was there before I started working on the article, I have no objection to removing it. Gramix13 (talk) 02:43, 24 May 2025 (UTC)[reply]

"Feel free to improve by adding citations or revert for the problems related to WP:CALC". What I meant here is that if the mathematical example is involved in a calculation like arithmetic, you don't have to add a citation. For the proofs, I think you might need a source; I will talk to the professionals. Dedhert.Jr (talk) 05:07, 26 May 2025 (UTC)[reply]

Update. See this recent discussion. Dedhert.Jr (talk) 05:56, 26 May 2025 (UTC)[reply]

Yeah so that seems to follow from what I have read from WP:CALC. While I think the result is important in establishing the relation between the size of the kernel and how the homomorphism "connects" elements in the domain when mapped to the image, I don't really think the proof itself is illustrative in demonstrating that result other that maybe showing that connection for group-like structures, so I will remove it from the article unless a good reason later comes up for including it. Gramix13 (talk) 07:08, 26 May 2025 (UTC)[reply]