List of common semigroups (which are not groups)?
Is there a list of common finite semigroups? I'm assuming a sensible classification as one has with finite simple groups is impossible, but it would be nice to see a list of common small examples, however incomplete.
For example, if one is testing a simple proposition about finite semigroups against some examples, what does one start with?
big-list semigroups
edited Nov 30 at 23:28
Shaun
8,686113680
asked Oct 29 '17 at 2:59
ThoralfSkolem
1,133615
add a comment |
Is there a list of common finite semigroups? I'm assuming a sensible classification as one has with finite simple groups is impossible, but it would be nice to see a list of common small examples, however incomplete.
For example, if one is testing a simple proposition about finite semigroups against some examples, what does one start with?
big-list semigroups
edited Nov 30 at 23:28
Shaun
8,686113680
asked Oct 29 '17 at 2:59
ThoralfSkolem
1,133615
en.wikipedia.org/wiki/Semigroup_with_two_elements and en.wikipedia.org/wiki/Semigroup_with_three_elements give 23 inequivalent semigroups of order 2 and 3, two of which are themselves groups. Important ones include boolean AND and multiplication on {-1, 0, 1}
– Dan Uznanski
Oct 29 '17 at 3:11
Which classification of groups do you have in mind?
– José Carlos Santos
Oct 29 '17 at 11:58
I probably meant the classification of finite simple groups.
– ThoralfSkolem
Oct 30 '17 at 16:03
This might interest you.
– Shaun
Nov 30 at 23:20
add a comment |
Is there a list of common finite semigroups? I'm assuming a sensible classification as one has with finite simple groups is impossible, but it would be nice to see a list of common small examples, however incomplete.
For example, if one is testing a simple proposition about finite semigroups against some examples, what does one start with?
big-list semigroups
edited Nov 30 at 23:28
Shaun
8,686113680
asked Oct 29 '17 at 2:59
ThoralfSkolem
1,133615
Is there a list of common finite semigroups? I'm assuming a sensible classification as one has with finite simple groups is impossible, but it would be nice to see a list of common small examples, however incomplete.
For example, if one is testing a simple proposition about finite semigroups against some examples, what does one start with?
big-list semigroups
big-list semigroups
edited Nov 30 at 23:28
Shaun
8,686113680
asked Oct 29 '17 at 2:59
ThoralfSkolem
1,133615
edited Nov 30 at 23:28
Shaun
8,686113680
asked Oct 29 '17 at 2:59
ThoralfSkolem
1,133615
edited Nov 30 at 23:28
Shaun
8,686113680
edited Nov 30 at 23:28
Shaun
8,686113680
edited Nov 30 at 23:28
Shaun
8,686113680
8,686113680
asked Oct 29 '17 at 2:59
ThoralfSkolem
1,133615
asked Oct 29 '17 at 2:59
ThoralfSkolem
1,133615
asked Oct 29 '17 at 2:59
ThoralfSkolem
1,133615
1,133615
en.wikipedia.org/wiki/Semigroup_with_two_elements and en.wikipedia.org/wiki/Semigroup_with_three_elements give 23 inequivalent semigroups of order 2 and 3, two of which are themselves groups. Important ones include boolean AND and multiplication on {-1, 0, 1}
– Dan Uznanski
Oct 29 '17 at 3:11
Which classification of groups do you have in mind?
– José Carlos Santos
Oct 29 '17 at 11:58
I probably meant the classification of finite simple groups.
– ThoralfSkolem
Oct 30 '17 at 16:03
This might interest you.
– Shaun
Nov 30 at 23:20
add a comment |
en.wikipedia.org/wiki/Semigroup_with_two_elements and en.wikipedia.org/wiki/Semigroup_with_three_elements give 23 inequivalent semigroups of order 2 and 3, two of which are themselves groups. Important ones include boolean AND and multiplication on {-1, 0, 1}
– Dan Uznanski
Oct 29 '17 at 3:11
Which classification of groups do you have in mind?
– José Carlos Santos
Oct 29 '17 at 11:58
I probably meant the classification of finite simple groups.
– ThoralfSkolem
Oct 30 '17 at 16:03
This might interest you.
– Shaun
Nov 30 at 23:20
en.wikipedia.org/wiki/Semigroup_with_two_elements and en.wikipedia.org/wiki/Semigroup_with_three_elements give 23 inequivalent semigroups of order 2 and 3, two of which are themselves groups. Important ones include boolean AND and multiplication on {-1, 0, 1}
– Dan Uznanski
Oct 29 '17 at 3:11
en.wikipedia.org/wiki/Semigroup_with_two_elements and en.wikipedia.org/wiki/Semigroup_with_three_elements give 23 inequivalent semigroups of order 2 and 3, two of which are themselves groups. Important ones include boolean AND and multiplication on {-1, 0, 1}
– Dan Uznanski
Oct 29 '17 at 3:11
Which classification of groups do you have in mind?
– José Carlos Santos
Oct 29 '17 at 11:58
Which classification of groups do you have in mind?
– José Carlos Santos
Oct 29 '17 at 11:58
I probably meant the classification of finite simple groups.
– ThoralfSkolem
Oct 30 '17 at 16:03
I probably meant the classification of finite simple groups.
– ThoralfSkolem
Oct 30 '17 at 16:03
This might interest you.
– Shaun
Nov 30 at 23:20
This might interest you.
– Shaun
Nov 30 at 23:20
add a comment |
1 Answer
1
active
oldest
votes
Common semigroups of low order which are not groups.
The semigroup $N_2 = {a,0}$ where $0$ is a zero and $a^2 = 0$.
The monoid $U_1 = {1, 0}$ under usual multiplication of integers.
Let $I$ and $J$ be two nonempty sets. Define an operation on $I times J$ by setting, for every $(i,j), (i',j') in I times J$,
$$
(i,j)(i',j') = (i, j')
$$
This defines a semigroup of order $|I||J|$, usually denoted by $B(I, J)$.
Second part of your question.
Let $B_2$ be the set of $2 times 2$ matrices with $0$-$1$ entries and at most one nonzero entry. Equipped with the usual multiplication of matrices, $B_2$ is a semigroup.
$$
B_2 = left{
begin{pmatrix} 1&0 \ 0&0 end{pmatrix},
begin{pmatrix} 0&1 \ 0&0 end{pmatrix},
begin{pmatrix} 0&0 \ 1&0 end{pmatrix},
begin{pmatrix} 0&0 \ 0&1 end{pmatrix},
begin{pmatrix} 0&0 \ 0&0 end{pmatrix}
right}
$$
Setting $a=left(
begin{smallmatrix}
0 1\
0 0
end{smallmatrix}
right)
$
and
$
b=left(
begin{smallmatrix}
0 0\
1 0
end{smallmatrix}
right)
$, one gets
$ab=left(
begin{smallmatrix}
1 0\
0 0
end{smallmatrix}
right)
$,
$ba=left(
begin{smallmatrix}
0 0\
0 1
end{smallmatrix}
right)
$
and
$0=left(
begin{smallmatrix}
0 0\
0 0
end{smallmatrix}
right)$. Thus $B_2 = {a, b, ab, ba, 0}$. Furthermore, the relations $aa=bb=0$, $aba=a$ and $bab=b$ suffice to recover completely the multiplication in $B_2$. If one adds the identity matrix, one gets the 6-element monoid $B^1_2$. The semigroups $B_2$ and $B_2^1$ are nicknamed the universal counterexamples because they provide many counterexamples in semigroup theory.
answered Oct 29 '17 at 11:55
J.-E. Pin
18.3k21754
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2494451%2flist-of-common-semigroups-which-are-not-groups%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Common semigroups of low order which are not groups.
The semigroup $N_2 = {a,0}$ where $0$ is a zero and $a^2 = 0$.
The monoid $U_1 = {1, 0}$ under usual multiplication of integers.
Let $I$ and $J$ be two nonempty sets. Define an operation on $I times J$ by setting, for every $(i,j), (i',j') in I times J$,
$$
(i,j)(i',j') = (i, j')
$$
This defines a semigroup of order $|I||J|$, usually denoted by $B(I, J)$.
Second part of your question.
Let $B_2$ be the set of $2 times 2$ matrices with $0$-$1$ entries and at most one nonzero entry. Equipped with the usual multiplication of matrices, $B_2$ is a semigroup.
$$
B_2 = left{
begin{pmatrix} 1&0 \ 0&0 end{pmatrix},
begin{pmatrix} 0&1 \ 0&0 end{pmatrix},
begin{pmatrix} 0&0 \ 1&0 end{pmatrix},
begin{pmatrix} 0&0 \ 0&1 end{pmatrix},
begin{pmatrix} 0&0 \ 0&0 end{pmatrix}
right}
$$
Setting $a=left(
begin{smallmatrix}
0 1\
0 0
end{smallmatrix}
right)
$
and
$
b=left(
begin{smallmatrix}
0 0\
1 0
end{smallmatrix}
right)
$, one gets
$ab=left(
begin{smallmatrix}
1 0\
0 0
end{smallmatrix}
right)
$,
$ba=left(
begin{smallmatrix}
0 0\
0 1
end{smallmatrix}
right)
$
and
$0=left(
begin{smallmatrix}
0 0\
0 0
end{smallmatrix}
right)$. Thus $B_2 = {a, b, ab, ba, 0}$. Furthermore, the relations $aa=bb=0$, $aba=a$ and $bab=b$ suffice to recover completely the multiplication in $B_2$. If one adds the identity matrix, one gets the 6-element monoid $B^1_2$. The semigroups $B_2$ and $B_2^1$ are nicknamed the universal counterexamples because they provide many counterexamples in semigroup theory.
answered Oct 29 '17 at 11:55
J.-E. Pin
18.3k21754
add a comment |
Common semigroups of low order which are not groups.
The semigroup $N_2 = {a,0}$ where $0$ is a zero and $a^2 = 0$.
The monoid $U_1 = {1, 0}$ under usual multiplication of integers.
Let $I$ and $J$ be two nonempty sets. Define an operation on $I times J$ by setting, for every $(i,j), (i',j') in I times J$,
$$
(i,j)(i',j') = (i, j')
$$
This defines a semigroup of order $|I||J|$, usually denoted by $B(I, J)$.
Second part of your question.
Let $B_2$ be the set of $2 times 2$ matrices with $0$-$1$ entries and at most one nonzero entry. Equipped with the usual multiplication of matrices, $B_2$ is a semigroup.
$$
B_2 = left{
begin{pmatrix} 1&0 \ 0&0 end{pmatrix},
begin{pmatrix} 0&1 \ 0&0 end{pmatrix},
begin{pmatrix} 0&0 \ 1&0 end{pmatrix},
begin{pmatrix} 0&0 \ 0&1 end{pmatrix},
begin{pmatrix} 0&0 \ 0&0 end{pmatrix}
right}
$$
Setting $a=left(
begin{smallmatrix}
0 1\
0 0
end{smallmatrix}
right)
$
and
$
b=left(
begin{smallmatrix}
0 0\
1 0
end{smallmatrix}
right)
$, one gets
$ab=left(
begin{smallmatrix}
1 0\
0 0
end{smallmatrix}
right)
$,
$ba=left(
begin{smallmatrix}
0 0\
0 1
end{smallmatrix}
right)
$
and
$0=left(
begin{smallmatrix}
0 0\
0 0
end{smallmatrix}
right)$. Thus $B_2 = {a, b, ab, ba, 0}$. Furthermore, the relations $aa=bb=0$, $aba=a$ and $bab=b$ suffice to recover completely the multiplication in $B_2$. If one adds the identity matrix, one gets the 6-element monoid $B^1_2$. The semigroups $B_2$ and $B_2^1$ are nicknamed the universal counterexamples because they provide many counterexamples in semigroup theory.
answered Oct 29 '17 at 11:55
J.-E. Pin
18.3k21754
add a comment |
Common semigroups of low order which are not groups.
The semigroup $N_2 = {a,0}$ where $0$ is a zero and $a^2 = 0$.
The monoid $U_1 = {1, 0}$ under usual multiplication of integers.
Let $I$ and $J$ be two nonempty sets. Define an operation on $I times J$ by setting, for every $(i,j), (i',j') in I times J$,
$$
(i,j)(i',j') = (i, j')
$$
This defines a semigroup of order $|I||J|$, usually denoted by $B(I, J)$.
Second part of your question.
Let $B_2$ be the set of $2 times 2$ matrices with $0$-$1$ entries and at most one nonzero entry. Equipped with the usual multiplication of matrices, $B_2$ is a semigroup.
$$
B_2 = left{
begin{pmatrix} 1&0 \ 0&0 end{pmatrix},
begin{pmatrix} 0&1 \ 0&0 end{pmatrix},
begin{pmatrix} 0&0 \ 1&0 end{pmatrix},
begin{pmatrix} 0&0 \ 0&1 end{pmatrix},
begin{pmatrix} 0&0 \ 0&0 end{pmatrix}
right}
$$
Setting $a=left(
begin{smallmatrix}
0 1\
0 0
end{smallmatrix}
right)
$
and
$
b=left(
begin{smallmatrix}
0 0\
1 0
end{smallmatrix}
right)
$, one gets
$ab=left(
begin{smallmatrix}
1 0\
0 0
end{smallmatrix}
right)
$,
$ba=left(
begin{smallmatrix}
0 0\
0 1
end{smallmatrix}
right)
$
and
$0=left(
begin{smallmatrix}
0 0\
0 0
end{smallmatrix}
right)$. Thus $B_2 = {a, b, ab, ba, 0}$. Furthermore, the relations $aa=bb=0$, $aba=a$ and $bab=b$ suffice to recover completely the multiplication in $B_2$. If one adds the identity matrix, one gets the 6-element monoid $B^1_2$. The semigroups $B_2$ and $B_2^1$ are nicknamed the universal counterexamples because they provide many counterexamples in semigroup theory.
answered Oct 29 '17 at 11:55
J.-E. Pin
18.3k21754
Common semigroups of low order which are not groups.
The semigroup $N_2 = {a,0}$ where $0$ is a zero and $a^2 = 0$.
The monoid $U_1 = {1, 0}$ under usual multiplication of integers.
Let $I$ and $J$ be two nonempty sets. Define an operation on $I times J$ by setting, for every $(i,j), (i',j') in I times J$,
$$
(i,j)(i',j') = (i, j')
$$
This defines a semigroup of order $|I||J|$, usually denoted by $B(I, J)$.
Second part of your question.
Let $B_2$ be the set of $2 times 2$ matrices with $0$-$1$ entries and at most one nonzero entry. Equipped with the usual multiplication of matrices, $B_2$ is a semigroup.
$$
B_2 = left{
begin{pmatrix} 1&0 \ 0&0 end{pmatrix},
begin{pmatrix} 0&1 \ 0&0 end{pmatrix},
begin{pmatrix} 0&0 \ 1&0 end{pmatrix},
begin{pmatrix} 0&0 \ 0&1 end{pmatrix},
begin{pmatrix} 0&0 \ 0&0 end{pmatrix}
right}
$$
Setting $a=left(
begin{smallmatrix}
0 1\
0 0
end{smallmatrix}
right)
$
and
$
b=left(
begin{smallmatrix}
0 0\
1 0
end{smallmatrix}
right)
$, one gets
$ab=left(
begin{smallmatrix}
1 0\
0 0
end{smallmatrix}
right)
$,
$ba=left(
begin{smallmatrix}
0 0\
0 1
end{smallmatrix}
right)
$
and
$0=left(
begin{smallmatrix}
0 0\
0 0
end{smallmatrix}
right)$. Thus $B_2 = {a, b, ab, ba, 0}$. Furthermore, the relations $aa=bb=0$, $aba=a$ and $bab=b$ suffice to recover completely the multiplication in $B_2$. If one adds the identity matrix, one gets the 6-element monoid $B^1_2$. The semigroups $B_2$ and $B_2^1$ are nicknamed the universal counterexamples because they provide many counterexamples in semigroup theory.
answered Oct 29 '17 at 11:55
J.-E. Pin
18.3k21754
answered Oct 29 '17 at 11:55
J.-E. Pin
18.3k21754
answered Oct 29 '17 at 11:55
J.-E. Pin
18.3k21754
answered Oct 29 '17 at 11:55
J.-E. Pin
18.3k21754
18.3k21754
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2494451%2flist-of-common-semigroups-which-are-not-groups%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
en.wikipedia.org/wiki/Semigroup_with_two_elements and en.wikipedia.org/wiki/Semigroup_with_three_elements give 23 inequivalent semigroups of order 2 and 3, two of which are themselves groups. Important ones include boolean AND and multiplication on {-1, 0, 1}
– Dan Uznanski
Oct 29 '17 at 3:11
Which classification of groups do you have in mind?
– José Carlos Santos
Oct 29 '17 at 11:58
I probably meant the classification of finite simple groups.
– ThoralfSkolem
Oct 30 '17 at 16:03
This might interest you.
– Shaun
Nov 30 at 23:20