# Talk:Magma (algebra)

## 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)

- 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

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

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

## 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

- 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

- 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:
- "Magma" is also to me much more common than "groupoid", for
**this**meaning. - 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)

- "Magma" is also to me much more common than "groupoid", for

tinyurl.com/uh3t here -- I learned "groupoid" in school, not "magma". Today on WikiPedia was my first sighting of "magma" with this meaning, as contrasted with melted basalt under the Earth's crust. I strongly feel a disambiguation page is needed. If somebody, as I did, searches for "groupoid", he should see *both* definitions, via diaambiguation page, rather than only the category-theory one which is there now. Move the category-theory one qualified name such as groupoid (category theory), and either have brief groupoid (algebra) that points to Magma, or just link to here directly from disambiguation page.

On the *Mathematics Subject Classification (2000)* the word *magma*
apparently never appears.
I believe that there is no doubt that MSC represents standard current mathematical
terminology, hence I think that the page should be renamed
*Groupoid (algebra)*.
Popopp 16:21, 13 February 2007 (UTC)

- As already pointed out, the problem with using groupoid (algebra) is that the categorical meaning can also be defined as an algebraic structure. So groupoid (algebra) is still ambiguous. -- Fropuff 18:39, 13 February 2007 (UTC)

- You are right; I was not familiar with the other notion of Groupoid, so I missed the main point that the other notion is algebraic, too. Anyway, I still do not see why the present notion should be called
*Magma*(see also below). Since the main difference between the two notions is that one is a partial algebraic structure and the present one is a total algebraic structure, I believe that the right name for the present one could be*Groupoid (total algebraic structure)*, while the former Groupoid page might be eventually (but not necessarily) moved to*Groupoid (partial algebraic structure)*. One might also consider the number of operations, hence call this one*Groupoid (one operation)*. In order to definitely check the status of the use of*magma*among mathematics, I have made a search within AMS mathscinet database; I searched the word*magma*anywhere in reviews of papers with Primary classification 20n02 (Sets with a single binary operation). The search has given two matches, while the search of the word*groupoid*has given 172 matches! Searching on Arxiv.org is not that easy, I looked at some results for*magma-anywhere*, and I have found one case in which it is used in the present sense, but I have found many other uses for it (for example, [1] uses it for still another kind of algebraic structure-four operations!). My conclusion is that*Magma*is much more unusual and even more confusing than*Groupoid (whatever)*. In my opinion, the only thing to do is to choose the most appropriate*whatever*. As I mentioned, I suggest*Groupoid (total algebraic structure)*. —The preceding unsigned comment was added by 82.84.201.129 (talk) 21:04, 13 February 2007 (UTC).

- You are right; I was not familiar with the other notion of Groupoid, so I missed the main point that the other notion is algebraic, too. Anyway, I still do not see why the present notion should be called

I have thougt a lot about the problem. One one side, it is just a matter of terminology, hence it is not very relevant, as far as the mathematical content of the entry is correct. On the other side, it might cause problems to someone wanting to study the notion both on wikipedia and on the mathematical literature, or simply just trying to learn and understand definitions. But what conviced me to talk again about the terminology is the consideration that wikipedia is not original research, and is not original research even in naming conventions. Hence we must adhere to well established conventions which, as I argued, in this case coincide with those used in the Mathematical Subject Classification (and to the use of mathematicians working in the field as well: as I mentioned mathscinet in Primary classification 20n02 gives 2 matches for magma and 172 matches for groupoid!). Because of the above reasons I propose to use *Groupoid (Set with a single binary operation)*, which is the name used in the MSC Classification. Please make me know if you do not agree.--Popopp 08:15, 2 May 2007 (UTC)

Well, this is very confusing for people ignorant of algebra (like me). Could we put a disambiguation in the template: Template:Group-like structures. Because the two books I am reading right now don't give inverses, neutral, etc... to groupoids. Tony (talk) 16:43, 17 September 2010 (UTC)

Whatever happened to moving this page? It sounds like it would be quite appropriate. I'd do it myself, but the most recent suggestion ("Groupoid (Set with a single binary operation)") sounds fairly awkward. However, I don't know ewnough about the topic to suggest a better alternative. I'll try finishing the article and seeing if I can think of something better by then. If someone smarter than me could do it instead or at least offer an alternative, it'd be appreciated. 71.199.190.190 (talk) 00:32, 3 September 2011 (UTC)

The current naming of Wikipedia is the best in practice. There are two very different kinds of objects: (1) small categories where every morphism is invertible and (2) sets with a binary operation. For (1) there is only one well-established name, 'groupoid', and for (2) there are two well-established names, 'magma' and 'groupoid'. So it is just better to avoid collision, and use 'magma' for (2) --- and of course, mention the alternate naming for it. --- 2011-11-27

## 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.

## (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)

## monad

according to [[2]] 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)

## free magma/binary tree automorphisms

It seems to me that a free magma is necessarily endowed with any operation that can be performed on a (rooted, complete) binary tree that leaves the binary tree rooted and complete. Right? In essence, magmas have well-defined, natural automorphisms. Yet, the discussion here, and corresponding discussion in the cat theory pages are notable in being silent on this matter. Is this ommission due to an incompleteness of the article, or is this due to some unwillingness or "incorrectness" of discussing this topic? Maybe this is an expression of "forgetfullness" in the cat theory sense? In which case, is there a category of "free magmas with their natural automorphisms"? linas 20:23, 11 February 2007 (UTC)

## division ?

It would be helpful to provide an information about what division is. Following the given link take to a page where the best one can find is the section on abstract algebra, where division is "defined" as multiplication with the inverse. Here it seems you talk about something else. What can it be? The best I can imagine is the possibility to solve uniquely(?) the equation a x=b for any a,b. Must this solution be unique? Or is it only the surjectivity of left-multiplication, for all elements? I think several definitions are possible, but even knowing semigroups, magmas, etc quite well, I believe that this notion is far from being standard and should be explained.— MFH:Talk 21:46, 12 February 2007 (UTC)

## Because magma is a relatively simple, intuitive structure, the notions of Galois connection, free functor could be at least mentioned in the Free magma section

Would it be worth to mention the (rather hard) concepts of Galois connection and Adjoint functor using the (intuitively simple) example of magma? Now, these concepts are demonstrated on the example of free group — but I think a free group is a rather hard concept, a free magma is much more familiar and didactic (overall usage and applications in computer science). I thought of the following modifications on section Free magma:

- A clear distinction between algebraic structure (typeset with fraktur) and set (typeset with Roman), especially carefully distinguishing structure vs its universe (otherwise, the very "flavor" of Galois connection and adjoint functor cannot be rendered well didactically. Even if we do not mention forgetful functor explicitly, but at least the notation could refer to its tacit usage implicitly.
- Adding at least a summarizing mentioning of these concepts (marked below with red).

Thus, the section Free magma would look like this:

A **free magma** on a set *X* is the "most general possible" magma generated by the set *X* (that is there are no relations or axioms imposed on the generators; see free object). It can be described, in terms familiar in computer science, as the magma of binary trees with leaves labeled by elements of *X*. The operation is that of joining trees at the root. It therefore has a foundational role in syntax.

A free magma has the universal property such that, if is a function from the set *X* to the universe *N* of any magma , then there is a unique extension of to a morphism of magmas

This property establishes a special relatedness among two sets and two algebraic structures, where *N* is gotten from by “forgetting” the structure imposed by the operation (forgetful functor), and is gotten from *X* by regarding it as generators of a [free] algebraic structure (free functor). This special interlinkedness (somewhat resembling to Galois connection) can be generalized to the notion of adjoint functor.

*See also*: free semigroup, free group, Hall set

## Algebraic structure of NAND and NOR functions?

Is there a specific name for the algebraic structure of the boolean NAND(sheffer) and Logical_NOR(peirce) functions? They satisfy no other "common" algebraic axioms except for being closed, unital and commutative. —Preceding unsigned comment added by 129.13.186.1 (talk) 21:58, 6 August 2008 (UTC)

## Are there associative, divisible magmas without identity elements and/or non-associative magmas with identity elements but not inverse elements?

Looking at the "Types of magmas" section of this article, in the Hexagonal-shaped image of two paths of successive additional membership creteria from a magma to a group (divisibility-identity-associativity and associativity-identity-inversibility, inversibility being equivilent to divisibility where there is identity and associativity, and even without associativilty if different left- and right inverses are allowed in the definition of inversibility), I have wondered if a path *of proper subsets* could be formed either with identity first or identity last (or both). Can a magma have an identity element without being either associative or even left- and right-inversible (divisible)? It would surprise me if a magma could have an indentity element without being associative as long as was divisible and without being divisible as long as it was associative but not if the magma was neither associative nor divisible. And can a magma be associative and divisible (an associative quasigroup or a divisible semigroup, or equivilently both a quasigroup and a semigroup) without also having an identity element and thus being a group? Are there 2 paths from a magma to a group forming a hexagon of subsets or 3! = 6 paths forming a cube of subsets? Finally, are there special names for magmas with identity elements (if they aren't all loops and/or monoids including groups) or quasigroups that are also semigroups (if they aren't all groups)? Thanks. Kevin Lamoreau (talk) 19:40, 6 September 2008 (UTC)

- "Can a magma have an identity element without being either associative or even left- and right-inversible (divisible)?" Yes: the octonions under mutliplication have an identity element (1), but are neither commutative nor associative. On the other hand, the octonions do have multiplicative inverses. Sullivan.t.j (talk) 20:50, 6 September 2008 (UTC)

- I could have phrased that question better. I knew about monoids like the octonions, put I was wondering, in that question if a magma could satisfy three conditions: (1) existence of an identity element; (2) lack of associtativity; and (3) lack of divisibility (and thus lack of inversability). My second question was if a magma could satisfy the following three conditions: (1) absence of an identity element; (2) associativity; and (3) divisibility (the opposite of the first case). My final question was what names there were if there were any for those sets (or rather the ones whose members could satisfy all of those criteria like associativity, identity, divisibility and inversability but
*only had to satisfy***identity**in the first case and**associativity and divisibility**in the second case. I wasn't concerned with whether or not a magma was commutative here. Thanks for your attempt to answer part of my question though. Kevin Lamoreau (talk) 00:11, 7 September 2008 (UTC)

- I could have phrased that question better. I knew about monoids like the octonions, put I was wondering, in that question if a magma could satisfy three conditions: (1) existence of an identity element; (2) lack of associtativity; and (3) lack of divisibility (and thus lack of inversability). My second question was if a magma could satisfy the following three conditions: (1) absence of an identity element; (2) associativity; and (3) divisibility (the opposite of the first case). My final question was what names there were if there were any for those sets (or rather the ones whose members could satisfy all of those criteria like associativity, identity, divisibility and inversability but

- Your first set of conditions is easily satisfied - fill in a 5x5 multiplication table at random, add an identity element, and you are very likely to have an example. The 3-element magma I posted here appears to be an example too, but all its elements have an inverse (which you seem to think is ruled out by the lack of divisibility - maybe I've misunderstood what you meant). Your second of set of conditions is impossible to satisfy, unless you count the empty semigroup.

## Is or is not a magma a groupoid?

In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure.

http://en.wikipedia.org/wiki/Magma_(algebra)#Types_of_magmas

Is or is not a magma a groupoid? In the section "Types of magmas", groupoid completely reverses the definition of a magma. —Preceding unsigned comment added by Fantadox (talk • contribs) 13:01, 30 November 2009 (UTC)

- If you're referring to the "Group-like structures" table, that table is using the other (category-theoretic) meaning of "groupoid". I'm inclined to remove the table on the grounds that it's needlessly confusing and doesn't really add anything to the article anyway. --Zundark (talk) 13:48, 30 November 2009 (UTC)

## "Inversibility"?

Is "inversibility" a word? Is it the one frequently used in this topic? What is wrong with "invertibility" — does it mean something else? Shreevatsa (talk) 05:27, 30 May 2010 (UTC)

- Note: I have changed the article to say "invertibility" for now, on the assumption that the old word was just a mistake (this article is the only one on Wikipedia that contained the word). If I'm wrong, I'd like to know. Shreevatsa (talk) 05:01, 31 May 2010 (UTC)

a simple solution to the terminology "groupoid" ambiguity is the name 'nonassociative semigroup' or even shorter 'pre-semigroup', where the prefix 'pre-' is an abbreviation for 'nonassociative' Doc at ut (talk) 17:42, 14 March 2011 (UTC)doc at ut

## 'Semicategory'?

Is 'semicategory' the standard term for a set with a partial associative binary operation? As the name for a set endowed with a total associative binary operation is called a 'semigroup', and when we remove the axiom of identity from the definition of a monoid, we do not call it a 'semimonoid'. Thus, calling its non-closed cousin a 'semicategory' is inconsistent nomenclature. I propose we use the term 'semigroupoid' here and mention any alternate names on the target page. Further supporting this suggestion is, a) the 'semicategory' link actually redirects to the 'semigroupoid' page anyway; b) on a lesser note, 'semigroupoid' is the preferred term for the Haskell implementation of this algebraic structure. MaxwellEdisonPhD (talk) 12:50, 14 September 2013 (UTC)