List of common semigroups (which are not groups)?












1












$begingroup$


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







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$endgroup$












  • $begingroup$
    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}
    $endgroup$
    – Dan Uznanski
    Oct 29 '17 at 3:11












  • $begingroup$
    Which classification of groups do you have in mind?
    $endgroup$
    – José Carlos Santos
    Oct 29 '17 at 11:58










  • $begingroup$
    I probably meant the classification of finite simple groups.
    $endgroup$
    – ThoralfSkolem
    Oct 30 '17 at 16:03










  • $begingroup$
    This might interest you.
    $endgroup$
    – Shaun
    Nov 30 '18 at 23:20
















1












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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}
    $endgroup$
    – Dan Uznanski
    Oct 29 '17 at 3:11












  • $begingroup$
    Which classification of groups do you have in mind?
    $endgroup$
    – José Carlos Santos
    Oct 29 '17 at 11:58










  • $begingroup$
    I probably meant the classification of finite simple groups.
    $endgroup$
    – ThoralfSkolem
    Oct 30 '17 at 16:03










  • $begingroup$
    This might interest you.
    $endgroup$
    – Shaun
    Nov 30 '18 at 23:20














1












1








1


1



$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 30 '18 at 23:28









Shaun

8,832113681




8,832113681










asked Oct 29 '17 at 2:59









ThoralfSkolemThoralfSkolem

1,133615




1,133615












  • $begingroup$
    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}
    $endgroup$
    – Dan Uznanski
    Oct 29 '17 at 3:11












  • $begingroup$
    Which classification of groups do you have in mind?
    $endgroup$
    – José Carlos Santos
    Oct 29 '17 at 11:58










  • $begingroup$
    I probably meant the classification of finite simple groups.
    $endgroup$
    – ThoralfSkolem
    Oct 30 '17 at 16:03










  • $begingroup$
    This might interest you.
    $endgroup$
    – Shaun
    Nov 30 '18 at 23:20


















  • $begingroup$
    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}
    $endgroup$
    – Dan Uznanski
    Oct 29 '17 at 3:11












  • $begingroup$
    Which classification of groups do you have in mind?
    $endgroup$
    – José Carlos Santos
    Oct 29 '17 at 11:58










  • $begingroup$
    I probably meant the classification of finite simple groups.
    $endgroup$
    – ThoralfSkolem
    Oct 30 '17 at 16:03










  • $begingroup$
    This might interest you.
    $endgroup$
    – Shaun
    Nov 30 '18 at 23:20
















$begingroup$
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}
$endgroup$
– Dan Uznanski
Oct 29 '17 at 3:11






$begingroup$
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}
$endgroup$
– Dan Uznanski
Oct 29 '17 at 3:11














$begingroup$
Which classification of groups do you have in mind?
$endgroup$
– José Carlos Santos
Oct 29 '17 at 11:58




$begingroup$
Which classification of groups do you have in mind?
$endgroup$
– José Carlos Santos
Oct 29 '17 at 11:58












$begingroup$
I probably meant the classification of finite simple groups.
$endgroup$
– ThoralfSkolem
Oct 30 '17 at 16:03




$begingroup$
I probably meant the classification of finite simple groups.
$endgroup$
– ThoralfSkolem
Oct 30 '17 at 16:03












$begingroup$
This might interest you.
$endgroup$
– Shaun
Nov 30 '18 at 23:20




$begingroup$
This might interest you.
$endgroup$
– Shaun
Nov 30 '18 at 23:20










1 Answer
1






active

oldest

votes


















3












$begingroup$

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.






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    3












    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$


















      3












      $begingroup$

      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.






      share|cite|improve this answer









      $endgroup$
















        3












        3








        3





        $begingroup$

        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.






        share|cite|improve this answer









        $endgroup$



        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Oct 29 '17 at 11:55









        J.-E. PinJ.-E. Pin

        18.3k21754




        18.3k21754






























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