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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:
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)
I've actually made that basic rather than simple, which is certainly POV or worse.
Charles Matthews 16:53, 27 Feb 2004 (UTC)
I'd like to suggest that the adjective "closed" be dropped from the definition entirely. If we define a magma to be a set M equipped with a "binary operation on M ", then there's really no chance of confusion (the key is the preposition "on".) The more general notion of a function M × M to N, where the codomain is a second set N, shouldn't be called a binary operation on M. The older literature may have used this terminology, but it's relatively rare to encounter it in contemporary writing. Moreover, I think it's confusing: closure is important when dealing with subobjects, but for the object itself, the adjective raises more questions than it answers (e.g., "why wouldn't a function M × M to M be closed? what would it even mean not to be?", etc.)
Peterlittig ( talk) 20:11, 30 April 2016 (UTC)
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)
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)
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)
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
Would it be better to say that quasigroups allow cancelation, rather than division? Division implies that there is an inverse for each element.
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)
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)
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 endomorphisms. 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)
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)
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:
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
-- 23:14, 27 February 2008 User:Physis
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)
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)
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)
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)
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
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)
The article says, "Magmas are not often studied as such". The lack of interesting theorems might be the reason. — Preceding unsigned comment added by 64.38.197.205 ( talk) 11:02, 25 July 2014 (UTC)
According to [3] Bourbaki called a magma what today is called a semigroup, i.e. having associativity. The source could be wrong though, but it was written by two French sounding names... JMP EAX ( talk) 12:15, 26 August 2014 (UTC)
According to [4] "magma" with the meaning given in this article was used [and probably introduced] by Serre. JMP EAX ( talk) 12:32, 26 August 2014 (UTC)
The article has a definition of a zeropotent magma that is not equivalent to nilpotent. I cannot find this term used at all in a Google search. Should we delete this definition? — Quondum 21:32, 11 January 2015 (UTC)
Hi !
I wonder if the definition of a magma requires the set to be non-empty. If so, i think it would be relevent to mention it in the article...
Best regards ! — Preceding unsigned comment added by 109.23.11.115 ( talk) 18:54, 8 March 2015 (UTC)
In the table labeled "Group-like structures", it has groups and the empty domain under the same category, and states that neither groups nor the empty domain require an identity element. Not a big deal, but does seem like an error. The empty domain should be given it's own row perhaps. — Preceding unsigned comment added by 2600:1700:C821:39D0:FD8C:2365:6D39:F3B2 ( talk) 20:31, 23 June 2022 (UTC)
What is the definition of homogeneous element in free magma? 213.6.145.233 ( talk) 09:19, 26 September 2022 (UTC)
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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:
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)
I've actually made that basic rather than simple, which is certainly POV or worse.
Charles Matthews 16:53, 27 Feb 2004 (UTC)
I'd like to suggest that the adjective "closed" be dropped from the definition entirely. If we define a magma to be a set M equipped with a "binary operation on M ", then there's really no chance of confusion (the key is the preposition "on".) The more general notion of a function M × M to N, where the codomain is a second set N, shouldn't be called a binary operation on M. The older literature may have used this terminology, but it's relatively rare to encounter it in contemporary writing. Moreover, I think it's confusing: closure is important when dealing with subobjects, but for the object itself, the adjective raises more questions than it answers (e.g., "why wouldn't a function M × M to M be closed? what would it even mean not to be?", etc.)
Peterlittig ( talk) 20:11, 30 April 2016 (UTC)
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)
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)
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)
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
Would it be better to say that quasigroups allow cancelation, rather than division? Division implies that there is an inverse for each element.
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)
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)
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 endomorphisms. 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)
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)
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:
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
-- 23:14, 27 February 2008 User:Physis
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)
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)
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)
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)
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
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)
The article says, "Magmas are not often studied as such". The lack of interesting theorems might be the reason. — Preceding unsigned comment added by 64.38.197.205 ( talk) 11:02, 25 July 2014 (UTC)
According to [3] Bourbaki called a magma what today is called a semigroup, i.e. having associativity. The source could be wrong though, but it was written by two French sounding names... JMP EAX ( talk) 12:15, 26 August 2014 (UTC)
According to [4] "magma" with the meaning given in this article was used [and probably introduced] by Serre. JMP EAX ( talk) 12:32, 26 August 2014 (UTC)
The article has a definition of a zeropotent magma that is not equivalent to nilpotent. I cannot find this term used at all in a Google search. Should we delete this definition? — Quondum 21:32, 11 January 2015 (UTC)
Hi !
I wonder if the definition of a magma requires the set to be non-empty. If so, i think it would be relevent to mention it in the article...
Best regards ! — Preceding unsigned comment added by 109.23.11.115 ( talk) 18:54, 8 March 2015 (UTC)
In the table labeled "Group-like structures", it has groups and the empty domain under the same category, and states that neither groups nor the empty domain require an identity element. Not a big deal, but does seem like an error. The empty domain should be given it's own row perhaps. — Preceding unsigned comment added by 2600:1700:C821:39D0:FD8C:2365:6D39:F3B2 ( talk) 20:31, 23 June 2022 (UTC)
What is the definition of homogeneous element in free magma? 213.6.145.233 ( talk) 09:19, 26 September 2022 (UTC)