# Who invented Monoid?

I was trying to find (and failed) the original author of either

• the concept of Monoid (set with binary associative operation and identity)
• the name (which sounds french ? and also Dioid (for what seems to be a semiring) is exclusively french wiki article)

Question:

Is there a text source to attribute the invention of either the name or concept of to a person or a group (pun intended) ?

• Actually, groups in the XIXth century were initially defined as subsets of (finite) symmetric groups with identity and closed under product. When von Dyck around 1880 (inspired by Cayley) defined groups abstractly he chose to force the existence of inverses, but I'm not sure this choice was immediately universal. For instance I've seen a 1915 paper by Andreoli (in Italian) about the monoid of self-maps of an infinite set, where he uses "group" for what is now called "monoid". – YCor Aug 13 at 18:08
• @YCor The condition on a set with binary isomorphism "$X$ is isomorphic to a set of permutations of an infinite set closed under multiplication and containing the identity" is actually relatively complicated to define intrinsically (it's a "right reversible cancellative monoid" in modern terminology, I believe). This is perhaps one reason that this perspective has fallen out of favor in modern mathematics? Of course with self-maps it's no problem. – Will Sawin Aug 13 at 19:27
• @WillSawin sure, I'm not blaming this choice of outcome for the word "group", but my point is that given the (well-developed) meaning of "group" in 1870, it was not obvious at all that it would eventually reach the modern definition, and could have reached the meaning of monoid; Andreoli (1915) does it and calls "complete group" what we call "group"— he seems to ignore the developments of the last 3 decades. – YCor Aug 13 at 19:53
• @YCor I didn't intend to disagree with what you wrote - this is just a thought I had upon reading your comment. – Will Sawin Aug 13 at 23:32
• So, you're looking for the identity of the term's inventor? *cough* – David Richerby Aug 14 at 8:02

The name "monoid" was first used in mathematics by Arthur Cayley [*] for a surface of order $$n$$ which has a multiple point of order $$n-1$$.

In the context of semigroups the name is due to Bourbaki [source, page 30]

It is also worth commenting on the related term monoid, meaning an associative magma with identity. This term is a little more recent than semigroup, and seems to originate with Bourbaki [**]. Before this, Birkhoff (1934) was using the term groupoid for an associative magma with identity.

Dov Tamari [source, page 1] argues that Bourbaki "probably intended to abolish the term semigroup for reasons of linguistic taste."

[*] A. Cayley, Second and Third Memoirs on Skew Surfaces, Otherwise Scrolls, Phil. Trans. (1863 and 1869).
[**] N. Bourbaki, Éléments de Mathématique, Algèbre, Hermann, Paris (1943): Chapter I, §2.

• Is there any connection to what we now know as monoids? – darij grinberg Aug 13 at 18:35
• monoidal surfaces are unrelated to monoidal semigroups, but this is where the term entered math. – Carlo Beenakker Aug 13 at 18:37
• More precisely Bourbaki (1942, p. 7): Un ensemble muni de la structure déterminée par une loi partout définie associative prend le nom de monoïde. Perhaps this was motivated by Eilenberg & Mac Lane’s upcoming A monoid is a category with one object ? (They started categories around 1942.) – Francois Ziegler Aug 13 at 21:18
• A third algebraic structure called a groupoid! – David Roberts Aug 13 at 21:25
• @FrancoisZiegler, thank you! Now I see why "monoid"!!! Just one object. – Wlod AA Aug 13 at 22:25