Mixing
Right zero in a finite semigroup
$begingroup$
Let $(M, cdot)$ be a finite semigroup such that
$$ x,yin Mwedge exists a,bin M:x=a⋅ywedge y=b⋅xRightarrow x=y. $$
Show that M contains at least one right absorbant element(or right zero).
Using that there exists an $x^n=x$ for $xin M$ because it is a finite semigroup I got that $(M, cdot)$ is a idempotent semigroup but I'm not sure it's correct.I don't know how to continue.
group-theory semigroups finite-semigroups
edited Jan 20 at 17:40
Dog_69
6441523
asked Jan 20 at 17:09
Eduard LaitinEduard Laitin
12
$endgroup$
add a comment |
$begingroup$
Let $(M, cdot)$ be a finite semigroup such that
$$ x,yin Mwedge exists a,bin M:x=a⋅ywedge y=b⋅xRightarrow x=y. $$
Show that M contains at least one right absorbant element(or right zero).
Using that there exists an $x^n=x$ for $xin M$ because it is a finite semigroup I got that $(M, cdot)$ is a idempotent semigroup but I'm not sure it's correct.I don't know how to continue.
group-theory semigroups finite-semigroups
edited Jan 20 at 17:40
Dog_69
6441523
asked Jan 20 at 17:09
Eduard LaitinEduard Laitin
12
$endgroup$
$begingroup$
What about a group where $a=b=e$? That doesn't have a zero element. That seems to satisfy your condition
$endgroup$
– Matt Samuel
Jan 20 at 17:34
$begingroup$
@MattSamuel Maybe OP's condition misses somethin like $aneq b$. I don't know.
$endgroup$
– Dog_69
Jan 20 at 17:40
$begingroup$
What do you mean by "Using that there exists an $x^n=x$ for $xin M$ because it is a finite semigroup"?
$endgroup$
– J.-E. Pin
Feb 5 at 10:51
add a comment |
$begingroup$
Let $(M, cdot)$ be a finite semigroup such that
$$ x,yin Mwedge exists a,bin M:x=a⋅ywedge y=b⋅xRightarrow x=y. $$
Show that M contains at least one right absorbant element(or right zero).
Using that there exists an $x^n=x$ for $xin M$ because it is a finite semigroup I got that $(M, cdot)$ is a idempotent semigroup but I'm not sure it's correct.I don't know how to continue.
group-theory semigroups finite-semigroups
edited Jan 20 at 17:40
Dog_69
6441523
asked Jan 20 at 17:09
Eduard LaitinEduard Laitin
12
$endgroup$
Let $(M, cdot)$ be a finite semigroup such that
$$ x,yin Mwedge exists a,bin M:x=a⋅ywedge y=b⋅xRightarrow x=y. $$
Show that M contains at least one right absorbant element(or right zero).
Using that there exists an $x^n=x$ for $xin M$ because it is a finite semigroup I got that $(M, cdot)$ is a idempotent semigroup but I'm not sure it's correct.I don't know how to continue.
group-theory semigroups finite-semigroups
group-theory semigroups finite-semigroups
edited Jan 20 at 17:40
Dog_69
6441523
asked Jan 20 at 17:09
Eduard LaitinEduard Laitin
12
edited Jan 20 at 17:40
Dog_69
6441523
asked Jan 20 at 17:09
Eduard LaitinEduard Laitin
12
edited Jan 20 at 17:40
Dog_69
6441523
edited Jan 20 at 17:40
Dog_69
6441523
edited Jan 20 at 17:40
Dog_69
6441523
6441523
asked Jan 20 at 17:09
Eduard LaitinEduard Laitin
12
asked Jan 20 at 17:09
Eduard LaitinEduard Laitin
12
asked Jan 20 at 17:09
Eduard LaitinEduard Laitin
12
12
$begingroup$
What about a group where $a=b=e$? That doesn't have a zero element. That seems to satisfy your condition
$endgroup$
– Matt Samuel
Jan 20 at 17:34
$begingroup$
@MattSamuel Maybe OP's condition misses somethin like $aneq b$. I don't know.
$endgroup$
– Dog_69
Jan 20 at 17:40
$begingroup$
What do you mean by "Using that there exists an $x^n=x$ for $xin M$ because it is a finite semigroup"?
$endgroup$
– J.-E. Pin
Feb 5 at 10:51
add a comment |
$begingroup$
What about a group where $a=b=e$? That doesn't have a zero element. That seems to satisfy your condition
$endgroup$
– Matt Samuel
Jan 20 at 17:34
$begingroup$
@MattSamuel Maybe OP's condition misses somethin like $aneq b$. I don't know.
$endgroup$
– Dog_69
Jan 20 at 17:40
$begingroup$
What do you mean by "Using that there exists an $x^n=x$ for $xin M$ because it is a finite semigroup"?
$endgroup$
– J.-E. Pin
Feb 5 at 10:51
$begingroup$
What about a group where $a=b=e$? That doesn't have a zero element. That seems to satisfy your condition
$endgroup$
– Matt Samuel
Jan 20 at 17:34
$begingroup$
What about a group where $a=b=e$? That doesn't have a zero element. That seems to satisfy your condition
$endgroup$
– Matt Samuel
Jan 20 at 17:34
$begingroup$
@MattSamuel Maybe OP's condition misses somethin like $aneq b$. I don't know.
$endgroup$
– Dog_69
Jan 20 at 17:40
$begingroup$
@MattSamuel Maybe OP's condition misses somethin like $aneq b$. I don't know.
$endgroup$
– Dog_69
Jan 20 at 17:40
$begingroup$
What do you mean by "Using that there exists an $x^n=x$ for $xin M$ because it is a finite semigroup"?
$endgroup$
– J.-E. Pin
Feb 5 at 10:51
$begingroup$
What do you mean by "Using that there exists an $x^n=x$ for $xin M$ because it is a finite semigroup"?
$endgroup$
– J.-E. Pin
Feb 5 at 10:51
add a comment |
1 Answer
1
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$begingroup$
Your condition means that $M$ is $mathcal{L}$-trivial, i.e. the Green's relation $mathcal{L}$ is the equality. Since $M$ is finite, it has a minimum ideal $I$, which is a completely simple semigroup. Moreover, since $M$ is $mathcal{L}$-trivial, $I$ is actually a right zero band, and all its elements are right zeroes.
answered Feb 5 at 11:03
J.-E. PinJ.-E. Pin
18.5k21754
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add a comment |
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1 Answer
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active
oldest
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1 Answer
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active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Your condition means that $M$ is $mathcal{L}$-trivial, i.e. the Green's relation $mathcal{L}$ is the equality. Since $M$ is finite, it has a minimum ideal $I$, which is a completely simple semigroup. Moreover, since $M$ is $mathcal{L}$-trivial, $I$ is actually a right zero band, and all its elements are right zeroes.
answered Feb 5 at 11:03
J.-E. PinJ.-E. Pin
18.5k21754
$endgroup$
add a comment |
$begingroup$
Your condition means that $M$ is $mathcal{L}$-trivial, i.e. the Green's relation $mathcal{L}$ is the equality. Since $M$ is finite, it has a minimum ideal $I$, which is a completely simple semigroup. Moreover, since $M$ is $mathcal{L}$-trivial, $I$ is actually a right zero band, and all its elements are right zeroes.
answered Feb 5 at 11:03
J.-E. PinJ.-E. Pin
18.5k21754
$endgroup$
add a comment |
$begingroup$
Your condition means that $M$ is $mathcal{L}$-trivial, i.e. the Green's relation $mathcal{L}$ is the equality. Since $M$ is finite, it has a minimum ideal $I$, which is a completely simple semigroup. Moreover, since $M$ is $mathcal{L}$-trivial, $I$ is actually a right zero band, and all its elements are right zeroes.
answered Feb 5 at 11:03
J.-E. PinJ.-E. Pin
18.5k21754
$endgroup$
Your condition means that $M$ is $mathcal{L}$-trivial, i.e. the Green's relation $mathcal{L}$ is the equality. Since $M$ is finite, it has a minimum ideal $I$, which is a completely simple semigroup. Moreover, since $M$ is $mathcal{L}$-trivial, $I$ is actually a right zero band, and all its elements are right zeroes.
answered Feb 5 at 11:03
J.-E. PinJ.-E. Pin
18.5k21754
answered Feb 5 at 11:03
J.-E. PinJ.-E. Pin
18.5k21754
answered Feb 5 at 11:03
J.-E. PinJ.-E. Pin
18.5k21754
answered Feb 5 at 11:03
J.-E. PinJ.-E. Pin
18.5k21754
18.5k21754
add a comment |
add a comment |
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$begingroup$
What about a group where $a=b=e$? That doesn't have a zero element. That seems to satisfy your condition
$endgroup$
– Matt Samuel
Jan 20 at 17:34
$begingroup$
@MattSamuel Maybe OP's condition misses somethin like $aneq b$. I don't know.
$endgroup$
– Dog_69
Jan 20 at 17:40
$begingroup$
What do you mean by "Using that there exists an $x^n=x$ for $xin M$ because it is a finite semigroup"?
$endgroup$
– J.-E. Pin
Feb 5 at 10:51