Who invented Monoid?Why “monoidal” transformation?Who invented the gamma function?Who invented the Morse-Bott-complex?Semiring naturally associated to any monoid?who invented projective space $mathbbP^n$?Who invented diagrammatic algebra?On functors preserving monoid objectsNaming in math: from red herrings to very long names
Who invented Monoid?
Why “monoidal” transformation?Who invented the gamma function?Who invented the Morse-Bott-complex?Semiring naturally associated to any monoid?who invented projective space $mathbbP^n$?Who invented diagrammatic algebra?On functors preserving monoid objectsNaming in math: from red herrings to very long names
$begingroup$
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 monoid to a person or a group (pun intended) ?
ct.category-theory soft-question ho.history-overview semigroups-and-monoids
edited 6 hours ago
Francois Ziegler
21.7k3 gold badges80 silver badges127 bronze badges
asked 10 hours ago
c69c69
1333 bronze badges
New contributor
c69 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
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 monoid to a person or a group (pun intended) ?
ct.category-theory soft-question ho.history-overview semigroups-and-monoids
edited 6 hours ago
Francois Ziegler
21.7k3 gold badges80 silver badges127 bronze badges
asked 10 hours ago
c69c69
1333 bronze badges
New contributor
c69 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
6
$begingroup$
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".
$endgroup$
– YCor
9 hours ago
$begingroup$
@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.
$endgroup$
– Will Sawin
8 hours ago
1
$begingroup$
@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.
$endgroup$
– YCor
8 hours ago
$begingroup$
@YCor I didn't intend to disagree with what you wrote - this is just a thought I had upon reading your comment.
$endgroup$
– Will Sawin
4 hours ago
add a comment |
$begingroup$
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 monoid to a person or a group (pun intended) ?
ct.category-theory soft-question ho.history-overview semigroups-and-monoids
edited 6 hours ago
Francois Ziegler
21.7k3 gold badges80 silver badges127 bronze badges
asked 10 hours ago
c69c69
1333 bronze badges
New contributor
c69 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
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 monoid to a person or a group (pun intended) ?
ct.category-theory soft-question ho.history-overview semigroups-and-monoids
ct.category-theory soft-question ho.history-overview semigroups-and-monoids
edited 6 hours ago
Francois Ziegler
21.7k3 gold badges80 silver badges127 bronze badges
asked 10 hours ago
c69c69
1333 bronze badges
New contributor
c69 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 6 hours ago
Francois Ziegler
21.7k3 gold badges80 silver badges127 bronze badges
asked 10 hours ago
c69c69
1333 bronze badges
New contributor
c69 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 6 hours ago
Francois Ziegler
21.7k3 gold badges80 silver badges127 bronze badges
edited 6 hours ago
Francois Ziegler
21.7k3 gold badges80 silver badges127 bronze badges
edited 6 hours ago
Francois Ziegler
21.7k3 gold badges80 silver badges127 bronze badges
21.7k3 gold badges80 silver badges127 bronze badges
asked 10 hours ago
c69c69
1333 bronze badges
New contributor
c69 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 10 hours ago
c69c69
1333 bronze badges
asked 10 hours ago
c69c69
1333 bronze badges
1333 bronze badges
New contributor
c69 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
c69 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
6
$begingroup$
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".
$endgroup$
– YCor
9 hours ago
$begingroup$
@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.
$endgroup$
– Will Sawin
8 hours ago
1
$begingroup$
@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.
$endgroup$
– YCor
8 hours ago
$begingroup$
@YCor I didn't intend to disagree with what you wrote - this is just a thought I had upon reading your comment.
$endgroup$
– Will Sawin
4 hours ago
add a comment |
6
$begingroup$
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".
$endgroup$
– YCor
9 hours ago
$begingroup$
@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.
$endgroup$
– Will Sawin
8 hours ago
1
$begingroup$
@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.
$endgroup$
– YCor
8 hours ago
$begingroup$
@YCor I didn't intend to disagree with what you wrote - this is just a thought I had upon reading your comment.
$endgroup$
– Will Sawin
4 hours ago
6
6
$begingroup$
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".
$endgroup$
– YCor
9 hours ago
$begingroup$
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".
$endgroup$
– YCor
9 hours ago
$begingroup$
@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.
$endgroup$
– Will Sawin
8 hours ago
$begingroup$
@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.
$endgroup$
– Will Sawin
8 hours ago
1
1
$begingroup$
@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.
$endgroup$
– YCor
8 hours ago
$begingroup$
@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.
$endgroup$
– YCor
8 hours ago
$begingroup$
@YCor I didn't intend to disagree with what you wrote - this is just a thought I had upon reading your comment.
$endgroup$
– Will Sawin
4 hours ago
$begingroup$
@YCor I didn't intend to disagree with what you wrote - this is just a thought I had upon reading your comment.
$endgroup$
– Will Sawin
4 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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.
answered 9 hours ago
Carlo BeenakkerCarlo Beenakker
86.2k9 gold badges205 silver badges311 bronze badges
$endgroup$
1
$begingroup$
Is there any connection to what we now know as monoids?
$endgroup$
– darij grinberg
9 hours ago
$begingroup$
This is a completely unrelated concept with the same name
$endgroup$
– Francesco Polizzi
9 hours ago
4
$begingroup$
monoidal surfaces are unrelated to monoidal semigroups, but this is where the term entered math.
$endgroup$
– Carlo Beenakker
9 hours ago
2
$begingroup$
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.)
$endgroup$
– Francois Ziegler
6 hours ago
1
$begingroup$
@FrancoisZiegler, thank you! Now I see why "monoid"!!! Just one object.
$endgroup$
– Wlod AA
5 hours ago
|
show 1 more comment
Your Answer
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1 Answer
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oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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.
answered 9 hours ago
Carlo BeenakkerCarlo Beenakker
86.2k9 gold badges205 silver badges311 bronze badges
$endgroup$
1
$begingroup$
Is there any connection to what we now know as monoids?
$endgroup$
– darij grinberg
9 hours ago
$begingroup$
This is a completely unrelated concept with the same name
$endgroup$
– Francesco Polizzi
9 hours ago
4
$begingroup$
monoidal surfaces are unrelated to monoidal semigroups, but this is where the term entered math.
$endgroup$
– Carlo Beenakker
9 hours ago
2
$begingroup$
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.)
$endgroup$
– Francois Ziegler
6 hours ago
1
$begingroup$
@FrancoisZiegler, thank you! Now I see why "monoid"!!! Just one object.
$endgroup$
– Wlod AA
5 hours ago
|
show 1 more comment
$begingroup$
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.
answered 9 hours ago
Carlo BeenakkerCarlo Beenakker
86.2k9 gold badges205 silver badges311 bronze badges
$endgroup$
1
$begingroup$
Is there any connection to what we now know as monoids?
$endgroup$
– darij grinberg
9 hours ago
$begingroup$
This is a completely unrelated concept with the same name
$endgroup$
– Francesco Polizzi
9 hours ago
4
$begingroup$
monoidal surfaces are unrelated to monoidal semigroups, but this is where the term entered math.
$endgroup$
– Carlo Beenakker
9 hours ago
2
$begingroup$
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.)
$endgroup$
– Francois Ziegler
6 hours ago
1
$begingroup$
@FrancoisZiegler, thank you! Now I see why "monoid"!!! Just one object.
$endgroup$
– Wlod AA
5 hours ago
|
show 1 more comment
$begingroup$
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.
answered 9 hours ago
Carlo BeenakkerCarlo Beenakker
86.2k9 gold badges205 silver badges311 bronze badges
$endgroup$
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.
answered 9 hours ago
Carlo BeenakkerCarlo Beenakker
86.2k9 gold badges205 silver badges311 bronze badges
edited 8 hours ago
answered 9 hours ago
Carlo BeenakkerCarlo Beenakker
86.2k9 gold badges205 silver badges311 bronze badges
answered 9 hours ago
Carlo BeenakkerCarlo Beenakker
86.2k9 gold badges205 silver badges311 bronze badges
answered 9 hours ago
Carlo BeenakkerCarlo Beenakker
86.2k9 gold badges205 silver badges311 bronze badges
86.2k9 gold badges205 silver badges311 bronze badges
1
$begingroup$
Is there any connection to what we now know as monoids?
$endgroup$
– darij grinberg
9 hours ago
$begingroup$
This is a completely unrelated concept with the same name
$endgroup$
– Francesco Polizzi
9 hours ago
4
$begingroup$
monoidal surfaces are unrelated to monoidal semigroups, but this is where the term entered math.
$endgroup$
– Carlo Beenakker
9 hours ago
2
$begingroup$
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.)
$endgroup$
– Francois Ziegler
6 hours ago
1
$begingroup$
@FrancoisZiegler, thank you! Now I see why "monoid"!!! Just one object.
$endgroup$
– Wlod AA
5 hours ago
|
show 1 more comment
1
$begingroup$
Is there any connection to what we now know as monoids?
$endgroup$
– darij grinberg
9 hours ago
$begingroup$
This is a completely unrelated concept with the same name
$endgroup$
– Francesco Polizzi
9 hours ago
4
$begingroup$
monoidal surfaces are unrelated to monoidal semigroups, but this is where the term entered math.
$endgroup$
– Carlo Beenakker
9 hours ago
2
$begingroup$
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.)
$endgroup$
– Francois Ziegler
6 hours ago
1
$begingroup$
@FrancoisZiegler, thank you! Now I see why "monoid"!!! Just one object.
$endgroup$
– Wlod AA
5 hours ago
1
1
$begingroup$
Is there any connection to what we now know as monoids?
$endgroup$
– darij grinberg
9 hours ago
$begingroup$
Is there any connection to what we now know as monoids?
$endgroup$
– darij grinberg
9 hours ago
$begingroup$
This is a completely unrelated concept with the same name
$endgroup$
– Francesco Polizzi
9 hours ago
$begingroup$
This is a completely unrelated concept with the same name
$endgroup$
– Francesco Polizzi
9 hours ago
4
4
$begingroup$
monoidal surfaces are unrelated to monoidal semigroups, but this is where the term entered math.
$endgroup$
– Carlo Beenakker
9 hours ago
$begingroup$
monoidal surfaces are unrelated to monoidal semigroups, but this is where the term entered math.
$endgroup$
– Carlo Beenakker
9 hours ago
2
2
$begingroup$
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.)
$endgroup$
– Francois Ziegler
6 hours ago
$begingroup$
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.)
$endgroup$
– Francois Ziegler
6 hours ago
1
1
$begingroup$
@FrancoisZiegler, thank you! Now I see why "monoid"!!! Just one object.
$endgroup$
– Wlod AA
5 hours ago
$begingroup$
@FrancoisZiegler, thank you! Now I see why "monoid"!!! Just one object.
$endgroup$
– Wlod AA
5 hours ago
|
show 1 more comment
c69 is a new contributor. Be nice, and check out our Code of Conduct.
c69 is a new contributor. Be nice, and check out our Code of Conduct.
c69 is a new contributor. Be nice, and check out our Code of Conduct.
c69 is a new contributor. Be nice, and check out our Code of Conduct.
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6
$begingroup$
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".
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– YCor
9 hours ago
$begingroup$
@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.
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– Will Sawin
8 hours ago
1
$begingroup$
@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.
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– YCor
8 hours ago
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@YCor I didn't intend to disagree with what you wrote - this is just a thought I had upon reading your comment.
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– Will Sawin
4 hours ago