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As I am not a native English speaker, I sometimes am bothered a little with the word "monoid", which is by definition a semigroup with identity. But why this terminology?

I searched some dictionaries (Longman for English, Larousse for Francais, Langenscheidts for Dentsch) but didn't find any result, and it seems to me that it is just a pronounciable word with certain mathematical meaning. So, where does it come from? Is there any etymological explanation? Who was the first mathematician who used it?

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    $\begingroup$ A couple of thoughts: a monoid is a structure with just ONE operation, and another name was needed other than group, semi-group, etc. Secondly, a monoid is, essentially, the same thing as a category with a SINGLE object. (Wikipedia) $\endgroup$
    – Old John
    Jun 11, 2012 at 12:54
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    $\begingroup$ @OldJohn Please put the categorical explanation in an answer :) $\endgroup$
    – rschwieb
    Jun 11, 2012 at 14:09

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For what it is worth, the Oxford English Dictionary traces monoid in this sense back to Chevalley's Fundamental Concept of Algebra published in 1956. Arthur Mattuck's review of the book in 1957 suggests that this use may be new, or at least new enough to be not in common mathematical parlance.

Edit:

  • Indeed, as recently as 1954 we've seen some use of the term "monoid" to mean a semigroup, not necessarily one with identity.
  • According to the OED again, the use of the word monoid in algebraic geometry (to denote "a surface which possesses a conical point of the highest possible order") dates back to 1866, and likely predates the use of the same term as semigroup with identity.
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    $\begingroup$ Much of the book is available on Google, including page 4, where Chevalley defines monoid. He doesn't give any motivation on that page. $\endgroup$ Jun 11, 2012 at 13:31
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    $\begingroup$ It seems the word "monoid" does not appear in Volume II of Clifford and Preston's seminal work "The Algebraic Theory of Semigroups" (1967), but it is in the late John Howie's text "An Introduction to Semigroup Theory" (1975). So, it seems to have gone from obscurity to common usage somewhere in these 8 years. $\endgroup$
    – user1729
    Jun 11, 2012 at 13:38
  • $\begingroup$ @Gerry: indeed. And I am not even sure whether Chevalley was the first one to put that term in print in its modern meaning, I'm just trying to give an indication of a possible answer (or some line to track) to the last question in the original post above. $\endgroup$ Jun 11, 2012 at 13:39
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    $\begingroup$ @user1729 I think we can back the date of common usage up to 1971, since MacLane's Categories for the Working Mathematician uses it casually. $\endgroup$
    – rschwieb
    Jun 11, 2012 at 14:10
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    $\begingroup$ @rschwieb: It is, perhaps, interesting that in the 1954 review, quoted in the post, it is Clifford reviewing and he pointedly does not use the word "monoid". So maybe he was against the word and didn't include it in his book for this reason, not because it wasn't in common usuage. $\endgroup$
    – user1729
    Jun 11, 2012 at 14:18
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If Chevalley was the first to popularize the term "monoid", then I can pretty confidently guess that it meant the structure of operators on a single type (i.e., a category with a single object). Note that Chevalley's second example (after the mandatory natural numbers) is the collection of mappings from a set to itself. His term for the monoid operation is "composition."

The term "groupoid" in the sense of a category with invertible arrows was already well-established. So, the use of "monoid" to mean a category of arrows on a single object seems quite natural.

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    $\begingroup$ (Much later) This seems like the strongest/simplest explanation, and probably the reason MacLane's book uses it freely. Linking "mono"<->"having identity" is tenuous at best. $\endgroup$
    – rschwieb
    Mar 17, 2017 at 14:43
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    $\begingroup$ (Much later again) @rschwieb: “mono” corresponding to “identity” doesn’t seem far-fetched at all — many terms about identities are derived from “unit” (“unital”, “unitary”, “unitor”…), and so someone wanting a term similar but slightly distinct might easily switch the Latin to Greek and coin “monoid”. That said, both these explanations seem plausible to me: I’d really like to see more historical evidence! $\endgroup$ Mar 23, 2020 at 18:35
  • $\begingroup$ @PeterLeFanuLumsdaine A groupoid is a category, and so all identity morphisms are automatically included. It seems to me no identity has been left behind, in this case. This seems to be at least one strike against the argument it has something to do with identities... Then notice nontrivial groupoids will definitely have more than one object. If you look at "group" as "a category with one object and all arrows invertible" then the two things above are natural generalizations, and their names seem to split the difference between the directions taken. $\endgroup$
    – rschwieb
    Mar 23, 2020 at 19:45
  • $\begingroup$ But it is not totally beyond the realm of possibility that what you say is true 🤷‍♂️ $\endgroup$
    – rschwieb
    Mar 23, 2020 at 19:49
  • $\begingroup$ Another strike against the theory is that the two concepts of "cardinality 1" and "identity of an operation" are really quite different despite both being symbolized as "1" in mathematical writing. The root "mono" refers to "a single thing" whereas the identity of an operation seems quite different. $\endgroup$
    – rschwieb
    Mar 23, 2020 at 19:52
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"mono" is a prefix meaning one, and a monoid is distinguished by having an identity element, which is frequently denoted by a one.

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  • $\begingroup$ Do you have a reference for that? It sounds like the most likely explanation for why the name monoid was chosen, but I'd like to see it from an authoritative source. $\endgroup$ Jun 11, 2012 at 13:09
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    $\begingroup$ Sorry, no. It's what I was taught many years ago, I simply believed the person who told me. $\endgroup$ Jun 11, 2012 at 13:16
  • $\begingroup$ It is here martinleslie.wordpress.com/2008/02/16/… but no reference for that either. $\endgroup$
    – GEdgar
    Jun 11, 2012 at 13:17
  • $\begingroup$ @GEdgar, yes, but as noted elsewhere on that page, that's wrong --- it says a monoid doesn't have an identity. $\endgroup$ Jun 11, 2012 at 13:24
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μόνος, -η, -ον (monos). In ancient greek means 'alone' or 'single. The 'monoid' in abstract algebra is a structure with a single binary operation. Single in ancient greek can be translated with the word 'monos', from which, the word 'monoid'

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    $\begingroup$ A group, a semigroup and a magma are also structures with a single binary operation. $\endgroup$ Jun 11, 2012 at 13:16
  • $\begingroup$ Some terminologies are originate from Greek or Latin. I guess the word 'monoid' is literally means $mono-identity$, i.e. there is only one identity in this structure. $\endgroup$
    – Popopo
    Jun 11, 2012 at 13:23
  • $\begingroup$ Or may be single refers to multiplication operation but the lack of inversion operation. $\endgroup$
    – palio
    Jun 11, 2012 at 13:25
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    $\begingroup$ Note to add that I downvoted this answer - if a reference can be provided that backs it up, I will gladly convert my downvote into an upvote. $\endgroup$ Jun 11, 2012 at 13:30

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