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Untitled

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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

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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

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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?

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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(ab) = 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)

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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

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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

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@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]

Universal algebra section

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Hello @D.Lazard, I wanted to clarify the Universal algebra section to make its topic more clear. This section is focused on homomorphism between algebraic structures and not on varieties.

Just as a quick summary of the two terms, algebraic structures are sets with a collection of operations on them. That collection might be indexed by a language, which associates each indexed operation with its arity. One example of a language would be where we have be a binary operation and denoting a constant. If we gave a set operations based on that language, that would give us operations and , where the super script indicates we are interpreting these operations with respect to the algebra . The interpretation gets important when we simultaneously discuss operations between algebraic structures, especially in the context of homomorphisms. This is what I am trying to define in the first paragraph of the section.

Varieties, in the context of universal algebra, are about classes of algebraic structures, more specifically those consisting of algebraic structure of a specific language that satisfy a collection of equations defined in that language. An equation like would be an example of one in the above language. This is not what the section is covering, and why there is a lack of mention towards any equations and why it seems like the definition here uses different... terminology.

Homomorphisms are defined for algebraic structures and not for varieties. They are also typically defined in the texts/sources I've read before varieties are defined, such as in Burris, Stanley; Sankappanavar, H.P. (2012) and McKenzie, Ralph; McNulty, George F.; Taylor, W. (1987). Varieties do satisfy the property that it is closed under homomorphic images of algebras within it, but to make sense of that one has to define homomorphism between algebras, which is what this section does.

As such, I don't feel that it is at all necessary to redefine this entire section in terms of varieties, the most natural way to handle homomorphisms would be to follow the way the sources handle the topic by doing it for algebraic structures, which is what I attempted to do in this section. I know that might not entirely resolve that section being unclear, but I do hope the above explanation help clarify your concerns in that section.

This edit I did try to make (which got reverted) attempted to clarify what I meant by the language in the algebra and separating it from the collection of operations on a particular algebra, I recommend rereading this diff and seeing if the way I wrote it there makes sense, and if not I am open to ways to make it more clear the way this is being defined.

One last thing I will mention is the equivalence relations is that I did notice is I didn't write down that should be the equivalence class of with respect to the equivalence relation , I'll go make an edit to clarify that wording, but if there's anything else that sounds confusing I'm open to suggestions on that front. Gramix13 (talk) 23:29, 9 August 2025 (UTC)[reply]

In universal algebra, algebraic structures are generally called algebras, and when a distinction is made between the two terms, algebraic structures may have axioms that are not universally quantified, while algebra may not. For example, fields are algebraic structures that are not algebras, because division by zero is not defined. A variety is a class of algebras that share the same signatures and the same axioms. Homomorphisms are defined only for algebra belonging to the same variety (for algebraic structures that are not algebras, homomorphisms can be defined, but with a slightly different definition: if is defined, then must be defined and one must have .
Instead of recalling this basic facts of universal algebra, the beginning of the section introduces another terminology. The use of "similar" instead of "belonging to the same variety" would be acceptable if clearly stated. The use of "of a similar type" as a synonym of "similar" is not acceptable. "Interpretation" is a term of model theory that has nothing to do here. I am unable to understand the meaning given to "indexed set" (clearly not indexed family and "langage" (clearly not formal language). In the whole, the first paragraph is a poor and confusing restatement of Variety (universal algebra)#Definition.
Another issue: You use for denoting . This is very confusing. would be acceptable but would need an explanation. Using is undoubtly the best way to express that and are equivalent for this equivalence relation. D.Lazard (talk) 09:02, 10 August 2025 (UTC)[reply]
Homomorphisms are defined only for algebra belonging to the same variety This is incorrect. The two sources I mentioned don't mention varieties in the definition of homomorphisms. I will quote each of their definitions to illustrate what I mean more directly.

Definition 6.1. Suppose and are two algebras of the same type . A mapping is called a homomorphism from to if

for each -ary in and each sequence from .

— Stanley Burris, H.P. Sankappanavar, "A Course in Universal Algebra", pp. 42-43

Now consider similar algebras and and let be an operation symbol of rank . A function from into is said to respect the interpretation of if and only if

for all .
DEFINITION 1.4. Let and be similar algebras. A function from into is called a homomorphism from into if and only if respects the interpretation of every operation symbol of .

— Ralph N. McKenzie, George F. McNulty, Walter F. Taylor, "Algebras, Lattices, Varieties", Volume I, p. 20

As you can plainly see, these just use the basic notion of an algebra, which doesn't assume any axioms at all, it only comes with operations based on its type.
Instead of recalling this basic facts of universal algebra, the beginning of the section introduces another terminology. I am precisely trying to recall those basic facts of universal algebra, by defining algebras themselves. I won't directly quote them here unless requested, but I would direct you to Definitions 2.1.1-2.1.3 in "A Course in Universal Algebra" (see page 23), and Definition 1.1 in "Algebras, Lattices, Varieties" (page 12).
The use of "of a similar type" as a synonym of "similar" is not acceptable. This is intentionally not a synonym, ALV uses the term "similarity type" to describe the rank function of an algebra (which maps each operation to its arity), and two algebras "are said to be similar if they have the same rank function." (page 13) This is precisely what I mean in that section.
"Interpretation" is a term of model theory that has nothing to do here. ALV does use the term interpretation when we talk about an operation for a specific algebra (page 12), and its the meaning I use here too.
I am unable to understand the meaning given to "indexed set" (clearly not indexed family and "langage" (clearly not formal language). So the indexed set is indeed an indexed family, its index set is what we call the language. But the language also has information about the arity of each operation it indexes. If two algebra use the same language, and we have an operation from each algebra indexed the same way in the language, then they have the same arity.
I do agree with how the notation of using directly as an equivalence relation might get confusing, so I will correct that in the article to match your suggestion. Gramix13 (talk) 15:45, 10 August 2025 (UTC)[reply]
The section is entitled § Universal algebra. So, it must use the standard terminology of universal algebra. This is not the case of the first paragraph. It does not matter whether some authors use another terminology (probably for pedagogical reasons or because they define kernels in a wider context than universal algebra). Doing otherwise would be confusing for readers who know already about universal algebra. The reference for terminology in universal algebra is Birkhoff's book.
Clearly, "having the same type" (or, if one want to be pedantic, "having similar types") is exactly the same as "belonging to the same variety".
By the way, for reaching the Good Article status, the article sould explain the generalization of the concept to categories :kernel (category theory) and kernel pair). D.Lazard (talk) 16:39, 10 August 2025 (UTC)[reply]
To the best of my knowledge, my definition of algebras attempts to use the standard terminology used in Universal Algebra. It might not be the most clear, and I can rewrite it with more detail to more closely follow the sources, but I am still trying to use the definition used in the two sources I've cited, and these sources clearly delineate between algebras and varieties. These sources define operations, followed by a set consisting of operations being indexed by a language (but these two sources differ on whether to define languages before or during the definition of an algebra).
I am open to reading Birkoff's book on the subject to see how he defines them, but I will need a more precise citation for the book since he has written numerous papers and books on algebra, and I do not want to misinterpret what book you are suggesting. I presume that these two sources were written afterwards, but while I am willing to mention in the article the definition given by Birkoff if it differs, I ultimately think the definition given should follow from what more recent sources give and if those definitions are more widely accepted by those in the field.
"Having the same type" is actually more broad and general than "belonging to the same variety". Groups, monoids, and semigroups all have the same type, if we consider their type to only consist of their singular binary operation, but each of them is in their own varieties, with the variety of groups strictly contained in the variety on monoids which is strictly contained in the variety of semigroups. If you insist on the variety terminology, what this section is handling is defining homomorphisms in the variety with no axioms, and homomorphisms in other varieties simply inherit this definition.
A more fundamental problem with defining homomorphisms for varieties is that one could also define it as a class of algebras closed under subalgebras, homomorphisms, and direct products. They are both equivalent via Birkoff's theorem. If someone were to use this definition of a variety (which is how the two sources I mentioned go about it), they would be lead to define homomorphisms for general algebras instead of in varieties because otherwise they might risk creating a circular argument where their definition of homomorphism between algebras in a variety must imply closure since it only accepts algebras in that variety.
I appreciate the suggestion for covering kernels in category theory, I will have to spend some time looking into them, but regardless of how this discussion on Universal Algebra goes, I will look into including them in this article. Gramix13 (talk) 17:16, 10 August 2025 (UTC)[reply]