Talk:Magma (algebra)

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Contents 1 Closed 2 Groupoid vs magma 3 division vs cancelation 4 (a*b)*c = (a*c)*(b*c) 5 monad

[edit] Closed

I wonder if the set is required to be closed under the operation—it's not clear from the article. For instance, would it be right to call Z a magma under the operation of taking the average of two integers, possibly yielding a non-integer?

Also, I think the opening sentence is misleading:

... a magma is a particularly simple kind of algebraic structure.

That's probably intended to mean that being a magma imposes little structure on a set, but one could also interpret it as: "If I can just show that my set here is a magma, why, then it must have a particularly simple algebraic structure!" Since highly complicated fields exist (and fields are magmas), that conclusion would be false.
—Herbee 16:24, 2004 Feb 27 (UTC)

By a binary operation on a set M, we mean a function from M×M to M. I think this is clear from the binary operation article. So a magma is necessarily closed. I agree that "simple" may be misleading - perhaps you can think of a better wording. (A field is not a magma; it's two interrelated magmas.) --Zundark 16:37, 27 Feb 2004 (UTC)

It's clear from you, Zundark; thanks for explaining. But it's not clear from the binary operation article, which points out that the term is sometimes used for any binary function, i.e. not necessarily from S × S to S.

A field not a magma? A field has a binary operation, so it fulfills the definition of a magma, doesn't it? Sure, it has another operation and more stuff, but that doesn't demagmafy it. Otherwise it would sound like "a bicicle doesn't have a wheel because it has two wheels!"

I wonder why they call it a "magma"? Next thing you know they'll rename a field to a "volcano", because of the magma in there...

—Herbee 22:11, 2004 Feb 27 (UTC)

I've actually made that basic rather than simple, which is certainly POV or worse.

Charles Matthews 16:53, 27 Feb 2004 (UTC)

[edit] Groupoid vs magma

We have this tendency to use weird terminology, and here's a good example. Groupoids are called magmas by almost nobody, basically just Bourbaki and the Magma computer algebra system. The people who actually study them call them groupoids. (The fact that every external link the page gives calls them groupoids should be a hint.) It's unfortunate that groupoid has two meanings, but it's not like we don't have plenty of tools to handle ambiguity for links. I suggest we move this to "groupoid (blah)", for suitable value for "blah".-- Walt Pohl 01:12, 16 Mar 2004 (UTC)

The title uses disambiguation anyway, so I don't see any objection to moving it. But I don't know what "blah" should be - "algebra" doesn't seem specific enough, as the other type of groupoid can also be considered as an algebraic object. --Zundark 17:22, 17 Dec 2004 (UTC)

I'd only ever heard of 'magma' - and it hadn't occurred to me that groupoid was ambiguous. Sorry, the other groupoid meaning here would be a horrid addition, except as a redirect. 'Magma' is good enough for Jean-Pierre Serre (I learned it from his lectures on Lie algebras), which makes it good enough for me.

Charles Matthews 17:30, 17 Dec 2004 (UTC)

But nobody who actually studies them calls them magmas. They call them groupoids. Magma is a Bourbaki-ism (so it's not surprising that Serre uses it) that never caught on among the actual practitioners of groupoids.

Well, I quibble at that. He's fussy about nomenclature, and not uncritical of Bourbaki. I have never come across a specialist in these things; apart that is from User:Deflog who added all these bitty definitions. But that is enough to undermine the idea that no one who works in the area says magma? Category theorists don't count? Well, it's anyway kind of obvious that category theorists would hate having 'groupoid' here. Charles Matthews

As for what to use for disambiguation, I don't know. Maybe 'groupoid (category theory)' for the other definition, and 'groupoid (algebra)' for the magma definition? I agree that the other kind of groupoids can be considered an algebraic object, so it's not perfect. Maybe 'groupoid (binary operation)'? -- Walt Pohl 20:34, 17 Dec 2004 (UTC)

I thought this over, and I still think we have no right to dictate which definition of groupoid is the right one. The term is used in two different senses in mathematics, and we should reflect that. -- Walt Pohl 06:32, 25 Jan 2005 (UTC)

Two comments:

1. "Magma" is also to me much more common than "groupoid", for this meaning.
2. I'm already worried about the "specification" "(algebra)" and just created a redirect for "Magma (mathematics)". Generalizing the principle that no such "specification" at all is added if there is no ambiguity, it seems most logical to me to add the name of the most largest possible "category" where no ambiguity exists. Else it is virtually impossible to guess for a user, what to add if (s)he searches the definition of an unknown term (and maybe does not even know what the precise meaning is). Thus:

• Why is this called Magma (algebra) and not Magma (mathematics) ?
• I really would regret addition of (binary operation) or even less intuitive things to "Magma", and I regret wherever an unnecessarily restrictive qualifier is added. (This seems b.t.w. especially common (almost exclusive) in this area of abstract algebra.)
• Just think about a term of everyday life or of a domain you're not common with, and what you would do if you were searching for its definition, lacking a disambiguation page. MFH 20:52, 17 Mar 2005 (UTC)

[edit] division vs cancelation

Would it be better to say that quasigroups allow cancelation, rather than division? Division implies that there is an inverse for each element.

[edit] (a*b)*c = (a*c)*(b*c)

In a completely unrealated subject I found that when I expressed a certian process as a binary operater it followed the identity (a*b)*c = (a*c)*(b*c). Is there any significance to this identity? --SurrealWarrior 19:50, 22 January 2006 (UTC)

It means that the operation is right distributive over itself. Any binary operation that is both idempotent and medial will have this property. For example, * could be the operation of taking the midpoint of two points in a real vector space. I don't know if this answers your question. --Zundark 20:51, 22 January 2006 (UTC)

Thank you! The process I was talking about is quite similar to taking the midpoint of two points in a real vector space, it is mixing two ideas to produce a third, for instance, mixing sports and player to produce a sports player.--SurrealWarrior 21:02, 22 January 2006 (UTC)

That sounds like you're looking for quandles. RandomP 19:04, 29 April 2006 (UTC)

[edit] monad

according to [[1]] a magma is synonymous for monad?

In dutch 'monade' is our name for that.

What do you think?

Evilbu 17:38, 16 February 2006 (UTC)

I think maybe the term magma isn't so standard. Some people call magmas groupoids (a usage which I don't condone, since groupoids should be categories with all invertible morphisms). Similarly, I guess some people call magmas monads, like the link you show. I don't like that usage either, because a monad should be monoid object in a monoidal category. Groupoids, monoids, and monads have identities and associativity, while magmas do not, so the different definitions are not very related. Of course, you can use any name you want, but using the name "groupoid" or "monad" for magma strikes me as unnecessarily ambiguous. -lethe ^{talk +} 00:34, 17 February 2006 (UTC)