Talk:Magma (algebra)

message: 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)
replies: text: 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)

text: 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)
replies: text: That sounds like you're looking for quandles. RandomP 19:04, 29 April 2006 (UTC)
thread_title:  (a*b)*c = (a*c)*(b*c) 

message: According to []
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)
replies: text: 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)
thread_title:  Monad 

message: 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)
replies: text: It's true that any magma has an automorphism group and that the elements of a free magma \mathbf{F}(X) on a generating set X can be identified with the rooted binary trees labeled with a member of X at each leaf (with the assignment of a «left» and «right» child node for each nonleaf node). I don't recall ever seeing a detailed account of automorphisms of free magmas and a short search just now returned nothing of note. Off the top of my head I'm not seeing any other automorphisms than those induced by permuting the elements of X, but there may well be other examples. It's worth noting that an automorphism of \mathbf{F}(X) permutes the set of those labeled trees, not the nodes of the trees themselves. If you're thinking that automorphisms of the free magmas are «natural» in the sense that they induce automorphisms of all other magmas that is incorrect. There is an automorphism of \mathbf{F}(\{x,y\}) which swaps x and y, yet the abelian group \mathbb{Z}_2\oplus\mathbb{Z}_3 (viewed as a magma) is a quotient of \mathbf{F}(\{x,y\}) generated by \{(1,0),(0,1)\} which does not have an automorphism swapping these two generators. That is, automorphisms of free magmas do not descend to automorphisms of all other magmas. The concept of «forgetfulness» in category theory doesn't apply here, since that is more about dropping structure as when we forget about the binary operation on a magma and only look at its underlying set. There is a category whose objects are the free magmas and whose morphisms are all the free magma automorphisms. It is a subcategory of the usual category whose objects are magmas and whose morphisms are magma homomorphisms. caterpillar_tree (talk) 23:17, 20 April 2020 (UTC)
replies: text: The original question was sloppy. The "natural maps" are the ones generated by the left-subtree map L: (a b)\mapsto a, the right subtree map R: (a b)\mapsto b, and the hyperbolic rotation \theta: ((a b) c)\mapsto (a (b c)).  Note that \theta is invertible. Obviously, these do not commute with the magma product itself; they are endomorphisms only in the sense that given an element of \mathbf{F}(X), they return another element of \mathbf{F}(X). They do preserve morphisms, in the sense that if f:M\to N is a morphism of magmas, these three maps all commute with f in that f\circ L = L\circ f and f\circ \theta = \theta\circ f  and etc. Because they commute, the universal property is conserved. Because the universal property is preserved, these are "natural". These are not the only ones, they generated an infinite set: for example, you can make some arbitrary number of left-right moves down the tree, and then apply \theta at that location, but keep the rest of the tree above that point unchanged. So there's an infinite number of these maps that are natural. The "rotation" is "hyperbolic" because binary trees embed into the modular group (which describes the hyperbolic rotations of the hyperbolic plane). And once you say these magic words, you've opened the flood-gates to all of riemann surfaces and number theory and what-not. So lets keep those flood-gates closed. Besides these, there are also also the tree-editing maps: e.g. take a bunch of left-right moves down the tree, take whatever you find there, and replace it with a different tree. So these are a raft of "self-evident" operations; seems like they belong in the article, right? 67.198.37.16 (talk) 01:21, 14 October 2020 (UTC)
replies: text: I'm not sure I'm following you here. If f\colon\mathbf{M}\to\mathbf{N} is a magma homomorphism then f\colon M\to N is a function which eats a single member of M at a time. You claim that f\circ L=L\circ f but as far as I can see L\colon F(X)\to F(X) is a function taking elements of the universe of the free magma (so in other words those rooted binary trees whose leaves are labeled with members of X) to other elements of the same set, which makes it looks like these two functions f and L cannot be composed with each other in any sensible way. Do you care to elaborate on what you meant by this? caterpillar_tree (talk) 20:35, 15 December 2021 (UTC)
thread_title:  Free magma/binary tree 
automorphisms
 endomorphisms 

message: 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)
thread_title:  Division ? 

message: 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 \mathfrak M_X 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 f:X\to N is a function from the set X to the universe N of any magma \mathfrak N, then there is a unique extension of f to a morphism of magmas f^\prime 
:f^\prime: \mathfrak M_X \to \mathfrak N. 
This property establishes a special relatedness among two sets and two algebraic structures, where N is gotten from \mathfrak N by “forgetting” the structure imposed by the operation (forgetful functor), and \mathfrak M_X 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
:The "somewhat resembling" is problematic. Anything adjoint to something forgetful "somewhat resembles" anything that's adjoint to something else. 67.198.37.16 (talk) 01:41, 14 October 2020 (UTC)
thread_title:  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 

message: 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)
thread_title:  Algebraic structure of NAND and NOR functions? 

message: In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure.
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http://en.wikipedia.org/wiki/Magma_(algebra)#Types_of_magmas
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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)
replies: text: 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)
thread_title:  Is or is not a magma a groupoid? 

message: 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)
replies: text: 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)
thread_title:  "Inversibility"? 

message: 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)
replies: text: Probably not or at least has multiple defs. See Talk:Semigroupoid. JMP EAX (talk) 11:42, 26 August 2014 (UTC)
thread_title:  'Semicategory'? 

message: 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)
thread_title:  Reason 

message: According to  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)
replies: text: Why not go straight to the source?  Bourbaki's Algebra I defines a "magma" on page 1 .  It is not required to be associative.  On page 4 is the definition of an "associative magma".  Deltahedron (talk) 16:41, 26 August 2014 (UTC)
thread_title:   Bourbaki 

message: 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)
replies: text: This definition in this article does not require the set to be nonempty. Some authors may exclude empty objects from consideration in their definitions, but I suspect that has more to do with not being bothered to think about it. I don't see any change needed in the article in this regard. —Quondum 21:02, 8 March 2015 (UTC)
thread_title:  Emptyness 

message: 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)
replies: text: It was added in this edit Feb 22, 2022 , changing from Inverse semigroup, note Inverse semigroups do not belong in the table; what fits, if anything, are possibly-empty groups. I don't know of a better wording, however.. I'll comment out the confusing row for now. Tom Ruen (talk) 21:46, 23 June 2022 (UTC)
thread_title:  Group doesn't possess an Identity element? 

message: What is the definition of homogeneous element in free magma? 213.6.145.233 (talk) 09:19, 26 September 2022 (UTC)
thread_title:  Homogeneous element