Talk:Monoid

message: could you give some examples related to computer ?
: Lists with the concatenation operation, and the empty list is a monoid  — Preceding unsigned comment added by 187.191.13.47 (talk) 23:24, 16 April 2013 (UTC)
thread_title:  Computer Examples

message: If an element in a monoid has an inverse is that inverse unique? Also why?
:Yes, associtivity guarantees that inverses, if they exist, are unique. Suppose b and c are both inverses of a. Then
::c = ce = c(ab) = (ca)b = eb = b
:So b = c. -- Fropuff 00:24, 13 January 2006 (UTC)
thread_title:  Uniqueness of inverses?

message: The term 'magma' is quite unnecessary in encyclopedic definition of monoid. Very few people use this word outside Bourbaki circles. The idea to insert an obscure term in a definition or an explanation - for students etc - is simply ridiculous.
Of course we need an article for magma, but it does not mean we should stick this word everywhere.
:I tend to agree. I'll remove it for now. -- Fropuff 03:23, 28 March 2006 (UTC)
thread_title:  Why magma in definition? 

message: The fourth example stated:
Any semigroup S may be turned into a monoid simply by adjoining an element e not in S and defining ee = e and es = s = se for all s ∈ S.  This can be done recursively (i.e. if S is monoid with identity e, one obtains a new monoid S ∪ {e′} with identity e′, and so on).
I removed the last sentence because by adjoining an identity e′ to a monoid with identity e simply gives e = e′ and the monoids are isomorphic.  
: No, that is false. Instead, given a monoid N with identity e (which is of course a semigroup), we can always embed N in a monoid M with identity e′ distinct from e; and if |N| is finite, this embedding cannot yield an isomorphism. (This is never the case if, e.g., we replace "monoid" with "group"). 71.198.111.245 20:05, 2 June 2007 (UTC)
replies: text: As far as regards the usefulness of the construction, I agree; except to demonstrate that monoid endomorphisms (which preserve the identity) and monoid homomorphisms aren't coincident. 71.198.111.245 20:05, 2 June 2007 (UTC)
thread_title:  Recursively adding an identity? 

message: Is it true that "every monoid contains a group" as written in the article?  For example, if Z+ forms a monoid over multiplication with identity one, what group is made with Z+?  I assume that to form a group, that set would need to be augmented with inverses.  Maybe the word "contains" is misleading me.  I can see how every group contains a monoid, but not vice-versa. - Simian1k 15:24, 13 April 2006 (UTC)
replies: text: I don't know why the article bothers to mention it, but it's certainly true: every monoid contains at least the trivial group {1}. --Zundark 17:42, 13 April 2006 (UTC)

text: We need to be a bit careful regarding what we mean by a "group contained in the monoid" here. If we mean "a subset of the monoid M which is a group w.r.t. the identity of M", then there is even a unique subset G of M which is maximal w.r.t. this property. On the other hand, consider the group C_7. As with any semigroup, it can be embedded in a monoid M = C_7 union {e} of order 8. The largest subset which is a group is of order 7; but the set of units has order 1. 71.198.111.245 20:05, 2 June 2007 (UTC)
thread_title:  Every monoid contains a group? 

message: These appear to be two different ways of saying the same thing: 1) A monoid satisfies all the axioms of a group with the exception of having inverses. A monoid with inverses is the same thing as a group; 2)If a monoid has the cancellation property and is finite, then it is in fact a group. drefty.mac 08:15, 28 October 2006 (UTC)
replies: text: They are not really the same thing. Cancellativity is (in general) a necessary but not a sufficient condition for the existence of inverses. So (1) does not immediately imply (2). And (2) is a statement only about finite monoids (it is false if you remove the hypothesis of finiteness) so it can't imply (1) in the general (possibly infinite) case. (Of course in a strict logical sense, any two statements which are provable imply each other, but you know what I mean.... :-). Best wishes, Cambyses 11:21, 23 November 2006 (UTC)
thread_title:  unify or clarify the following statements in the article 

message: The article states that there is a preordering if and only if there is a z such that x+z=y.  But the addition operator is not defined.  What am I missing?
—Preceding unsigned comment added by 82.235.223.112 (talk) 
:It's just the binary operation of the commutative monoid. I'll add a note to make this clearer. --Zundark 12:55, 8 May 2007 (UTC)
thread_title:  Addition operator 

message: One might want to add a pointer to the definition of Magma since it now exists in the Wikipedia. --Gdinolt 17:43, 21 May 2007 (UTC)
thread_title:  Add pointer to Magma Definition 

message: The article states:
:Equivalently, a submonoid is a subset N such that N=N*, where the superscript * is the Kleene star. For any subset N of M, the monoid N* is the smallest monoid that contains N.
First, strictly speaking, in the article on Wikipedia the Kleene star forms sets of strings of symbols from sets of symbols. Strings of symbols are not themselves symbols without some sort of (generally not injective) mapping. Of course, the mapping here is pretty obvious from context - if we already know what's going on, which should not be the point of view of the article IMO. A better link than to Kleene star explaining this idea might be free monoid or free object and some sort of discussion of presentation of a monoid similar to the article on presentation of a group.
Secondly, define (M,*) on the set {1,x} as: 1*1 = 1, 1*x = x*1 = x; x*x = x. M is a monoid. Let N = {x}. The semigroup (N,*\N} meets the requirement of being a monoid of order 1.  Is "the smallest monoid containing N" meant to be (N,*\N), or is it meant to be (M,*)? Equivalently, is {x} a submonoid of {1,x}? (Note that this confusion cannot arise with subgroups of a group). 
All of my comments here seem to revolve around the question of whether there is a preferred identity element 1 which is determined by the fact that we are working in the context of a /particular/ monoid M. Since monoids often are encountered as the multiplicative monoid of a ring with identity 1, I would expect that we usually map the empty string in N* to this 1, and not some other element of M; and that a submonoid N of M must contain 1; and that a group G embedded in M is assumed to be a submonoid of M. But I could be wrong! 71.198.111.245 20:05, 2 June 2007 (UTC)
thread_title:  Generators / Submonoids 

message: The sentence "This is the analogue in monoid theory of a (left) group action" seems a bit out of place in an article on monoids: what is "monoid theory" if not this article?   Maybe it was meant to be "group theory"?  Actually the article on group theory should refer to this article's treatment of actions, since a group is the same thing as a monoid whose left and right actions are all invertible; the fact that both left and right are invertible allows the element itself to have an inverse.  --Vaughan Pratt (talk) 09:07, 24 December 2008 (UTC)
thread_title: Actions

message: I can not find any reference to those. I guess that a find-first monoid is a monoid where, if an element is not neutral, it is left-absorbing. I doubt that "idempotent monoid" and "find-first monoid" coincide. An idempotent monoid is a monoid whose binary operation is idempotent. Every find-first monoid is idempotent.
Consider the monoid with the following table of the binary operation:
 	*	0	1
*	*	0	1
0	0	0	1
1	1	1	1
(In other words, this is the semigroup (\{0,1\},\lor), turned by the free functor into the monoid.) The monoid is idempotent. 0 is not neutral and is not left-absorbing. The monoid is not find-first. --Beroal (talk) 12:54, 15 January 2011 (UTC)
thread_title:  find-first monoid, find-last monoid 

message: Monoid congruence redirects here, but aside from its use in the section about presentations and the corresponding article on a presentation of a monoid, it is not mentioned. This is a problem, because no definition is given, so the redirect of monoid congruence to this article is really a dead end. Can someone who knows the definition of a monoid congruence please add the definition to this article? Furby100 (talk) 14:47, 26 April 2012 (UTC)
replies: text: I guess it is a particular case of Congruence relation. I changed that redirection. --Beroal (talk) 15:50, 4 May 2012 (UTC)
thread_title:  Monoid congruence definition is lacking 

message: The current article says "A submonoid of a monoid M is a subset N of M containing the unit element", but I don't see a definition for "unit element" that preceeds this.  I assume it means "identity element" in this context. Tashiro (talk) 01:19, 7 October 2012 (UTC)
replies: text: Fixed. — Quondum 05:38, 4 September 2013 (UTC)
thread_title: The "unit element"

message: I have changed the definition of a submonoid to match that of Jacobson, effectively reversing my earlier change that excluded the requirement that it include the identity of the enclosing monoid.  This is in line with a change of semantics from a subset that is a monoid under the monoid's binary operation to a subset that preserves the defined structure of the monoid, with the identity element being regarded as part of the defined structure (as in category theory, where it is regarded as a nullary operation).  This is compatible with the definition of a monoid homomorphism in this article, in the sense that the submonoids of a monoid are precisely the images of the monoid endomorphisms.
It may be noted that there are conflicting, but less notable, web definitions such as ,  and , which prompted my earlier edit. An article should give due weight to conflicting definitions. I leave it to others to determine whether this conflicting definition deserves more than the footnote that I included.  — Quondum 11:42, 26 September 2013 (UTC)
thread_title:  Definition of submonoid 

message: I like the new illustration :File:Exponentiation as monoid homomorphism.pdf added, but would like to suggest some tweaks to the diagram:
*Since we are dealing with the standard, known operations, the diagram should use the standard symbols, to wit: the monoid under multiplication should have its operation denoted ×, not *.
*Since (N,×) is a monoid and is a suitable codomain for the indicated mapping, there is no benefit in excluding 0. Excluding zero risks creating the impression that 0 must be excluded for some reason, and we should not do so, so I suggest including it in the multiplication table.
*It is sufficient to indicate the mapping on one axis only. I would suggest removing the arrows from the second axis, as the double mapping slows interpretation.
— Quondum 23:14, 9 October 2013 (UTC)
replies: text: Thanks, Jochen, for updating it as I suggested. I think that works very well. — Quondum 18:45, 10 October 2013 (UTC)
thread_title:  Suggested tweaks to new illustration 

message: The only difference between monoid and group is that in a group each element is required to have an inverse.  There is no "nontrivial theorem" needed to prove this; it is what the definitions currently on Wikipedia say. (The axioms are exactly the same, except for the existence of inverses.) Ebony Jackson (talk) 22:48, 7 December 2013 (UTC)
thread_title:  Monoid vs. group 

message: Subtle coincidence does not make the cut for encylopedic entries, I am going to edit the statement on identity element so that it states "There exists an element e in S such that for every element a in S, the equation  e • a = a • e   holds" as it makes no sense as written unless the typo was not as I will edit from and the correct statemeant is "There exists an element e in S such that for every element a in S, the equations e • a = a • e = a holes" . -Dirtclustit (talk) 15:03, 10 May 2015 (UTC)
thread_title:  Please be direct with your language 

message: In the section about monoid homomorphisms one can read "In contrast, a semigroup homomorphism between groups is always a group homomorphism,  as it necessarily preserves the identity."
Can you please explain why this is always the case?--Suitangi (talk) 16:14, 24 July 2018 (UTC)
thread_title:  Monoid homomorphisms 

message: The whole section on MapReduce is written in vague terms.  It's not clear what the monoid is.  Also it's not clear whether it is an actual application in the sense that the monoid terminology helps one understand it, or whether it is just something that turns out to be a monoid.  Unless a clearer description of the relevance can be given, I would suggest removing the section. Ebony Jackson (talk) 18:12, 2 January 2022 (UTC)
replies: text: I suggest to make it a subsection of "Monoids in computer science", since "Reduce" is a synonym for "fold", as far as I know. The text should be copy-edited; it could follow some easy-to-grasp application example from "big data", which probably can be found in the MapReduce article. I suspect that the work of Richard Bird, Lambert Meertens etc. on  algebraic methods in functional programming (Bird–Meertens formalism) is part of the theoretic foundation of MapReduce, without being recognized or acknowledged by most applied computer scientists. - Jochen Burghardt (talk) 20:07, 2 January 2022 (UTC)
thread_title:  MapReduce 

message: So far as I am aware, trace monoids are always infinite.  However, \{1, a, a^2\} with a^3 = a^2 is a finite (partially) commutative monoid which is not a trace monoid.  (I came to this article searching for a definition of an abstract trace monoid.)  I propose changing "A monoid for which the operation is commutative for some, but not all elements is a trace monoid; trace monoids commonly occur in the theory of concurrent computation" to "One type of monoid for which the operation is commutative for some, but not necessarily all, generators is the trace monoid; trace monoids commonly occur in the theory of concurrent computation".  Note that  for all monoids, so saying 'element' is not good. --RichardW57m (talk) 16:08, 4 April 2022 (UTC)
thread_title:  Trace Monoids