Definition and example of a bounded lattice?
$begingroup$
I understood what a lattice is and I read about some examples.
I understod what a semilattice is, what a complete lattice is but I don't know why I find some difficulties to get the concept of a bouned lattice.
Maybe it's because its very own name confuses me.
What does "bounded" mean?
If you look into wikipedia, it will just definite it as "a lattice with a greatest element and least element".
It would help to see some example like:
- A lattice that is bounded but it's not complete
- A lattice that is bounded and it's complete
- A lattice that isn't bounded and it's not complete
- A lattice that isn't bounded but it's complete
I know some examples are already discussed in other posts but they did not make it click for me, and, moreover, it would be much more intuitive to have all the cases in the same place so we can easily compare them.
lattice-orders
edited Dec 26 '18 at 10:38
Asaf Karagila♦
306k33438769
asked Dec 26 '18 at 10:30
Gabriele ScarlattiGabriele Scarlatti
370212
$endgroup$
add a comment |
$begingroup$
I understood what a lattice is and I read about some examples.
I understod what a semilattice is, what a complete lattice is but I don't know why I find some difficulties to get the concept of a bouned lattice.
Maybe it's because its very own name confuses me.
What does "bounded" mean?
If you look into wikipedia, it will just definite it as "a lattice with a greatest element and least element".
It would help to see some example like:
- A lattice that is bounded but it's not complete
- A lattice that is bounded and it's complete
- A lattice that isn't bounded and it's not complete
- A lattice that isn't bounded but it's complete
I know some examples are already discussed in other posts but they did not make it click for me, and, moreover, it would be much more intuitive to have all the cases in the same place so we can easily compare them.
lattice-orders
edited Dec 26 '18 at 10:38
Asaf Karagila♦
306k33438769
asked Dec 26 '18 at 10:30
Gabriele ScarlattiGabriele Scarlatti
370212
$endgroup$
add a comment |
$begingroup$
I understood what a lattice is and I read about some examples.
I understod what a semilattice is, what a complete lattice is but I don't know why I find some difficulties to get the concept of a bouned lattice.
Maybe it's because its very own name confuses me.
What does "bounded" mean?
If you look into wikipedia, it will just definite it as "a lattice with a greatest element and least element".
It would help to see some example like:
- A lattice that is bounded but it's not complete
- A lattice that is bounded and it's complete
- A lattice that isn't bounded and it's not complete
- A lattice that isn't bounded but it's complete
I know some examples are already discussed in other posts but they did not make it click for me, and, moreover, it would be much more intuitive to have all the cases in the same place so we can easily compare them.
lattice-orders
edited Dec 26 '18 at 10:38
Asaf Karagila♦
306k33438769
asked Dec 26 '18 at 10:30
Gabriele ScarlattiGabriele Scarlatti
370212
$endgroup$
I understood what a lattice is and I read about some examples.
I understod what a semilattice is, what a complete lattice is but I don't know why I find some difficulties to get the concept of a bouned lattice.
Maybe it's because its very own name confuses me.
What does "bounded" mean?
If you look into wikipedia, it will just definite it as "a lattice with a greatest element and least element".
It would help to see some example like:
- A lattice that is bounded but it's not complete
- A lattice that is bounded and it's complete
- A lattice that isn't bounded and it's not complete
- A lattice that isn't bounded but it's complete
I know some examples are already discussed in other posts but they did not make it click for me, and, moreover, it would be much more intuitive to have all the cases in the same place so we can easily compare them.
lattice-orders
lattice-orders
edited Dec 26 '18 at 10:38
Asaf Karagila♦
306k33438769
asked Dec 26 '18 at 10:30
Gabriele ScarlattiGabriele Scarlatti
370212
edited Dec 26 '18 at 10:38
Asaf Karagila♦
306k33438769
asked Dec 26 '18 at 10:30
Gabriele ScarlattiGabriele Scarlatti
370212
edited Dec 26 '18 at 10:38
Asaf Karagila♦
306k33438769
edited Dec 26 '18 at 10:38
Asaf Karagila♦
306k33438769
edited Dec 26 '18 at 10:38
Asaf Karagila♦
306k33438769
306k33438769
asked Dec 26 '18 at 10:30
Gabriele ScarlattiGabriele Scarlatti
370212
asked Dec 26 '18 at 10:30
Gabriele ScarlattiGabriele Scarlatti
370212
asked Dec 26 '18 at 10:30
Gabriele ScarlattiGabriele Scarlatti
370212
370212
add a comment |
add a comment |
1 Answer
1
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votes
$begingroup$
- A bounded, yet not complete lattice: take the set ${-1/n:ngeq 1} cup {1/n:ngeq 1}$ with the order inherited from $mathbb Q$. It is bounded, with least element $-1$ and greatest element $1$. Yet, it is not complete: the subset of negative numbers doesn't have a supremum within that set; likewise, the set of positive numbers doesn't have an infimum.
- Bounded and complete: just take any powerset lattice.
- Neither bounded nor complete: take the natural numbers with the usual order.
- Not bounded but complete: there is no such thing. If a lattice $L$ is complete, then it is bounded by $0_L = bigwedge L$ and $1_L = bigvee L$.
answered Dec 26 '18 at 10:58
amrsaamrsa
3,7852618
$endgroup$
add a comment |
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1 Answer
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active
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$begingroup$
- A bounded, yet not complete lattice: take the set ${-1/n:ngeq 1} cup {1/n:ngeq 1}$ with the order inherited from $mathbb Q$. It is bounded, with least element $-1$ and greatest element $1$. Yet, it is not complete: the subset of negative numbers doesn't have a supremum within that set; likewise, the set of positive numbers doesn't have an infimum.
- Bounded and complete: just take any powerset lattice.
- Neither bounded nor complete: take the natural numbers with the usual order.
- Not bounded but complete: there is no such thing. If a lattice $L$ is complete, then it is bounded by $0_L = bigwedge L$ and $1_L = bigvee L$.
answered Dec 26 '18 at 10:58
amrsaamrsa
3,7852618
$endgroup$
add a comment |
$begingroup$
- A bounded, yet not complete lattice: take the set ${-1/n:ngeq 1} cup {1/n:ngeq 1}$ with the order inherited from $mathbb Q$. It is bounded, with least element $-1$ and greatest element $1$. Yet, it is not complete: the subset of negative numbers doesn't have a supremum within that set; likewise, the set of positive numbers doesn't have an infimum.
- Bounded and complete: just take any powerset lattice.
- Neither bounded nor complete: take the natural numbers with the usual order.
- Not bounded but complete: there is no such thing. If a lattice $L$ is complete, then it is bounded by $0_L = bigwedge L$ and $1_L = bigvee L$.
answered Dec 26 '18 at 10:58
amrsaamrsa
3,7852618
$endgroup$
add a comment |
$begingroup$
- A bounded, yet not complete lattice: take the set ${-1/n:ngeq 1} cup {1/n:ngeq 1}$ with the order inherited from $mathbb Q$. It is bounded, with least element $-1$ and greatest element $1$. Yet, it is not complete: the subset of negative numbers doesn't have a supremum within that set; likewise, the set of positive numbers doesn't have an infimum.
- Bounded and complete: just take any powerset lattice.
- Neither bounded nor complete: take the natural numbers with the usual order.
- Not bounded but complete: there is no such thing. If a lattice $L$ is complete, then it is bounded by $0_L = bigwedge L$ and $1_L = bigvee L$.
answered Dec 26 '18 at 10:58
amrsaamrsa
3,7852618
$endgroup$
- A bounded, yet not complete lattice: take the set ${-1/n:ngeq 1} cup {1/n:ngeq 1}$ with the order inherited from $mathbb Q$. It is bounded, with least element $-1$ and greatest element $1$. Yet, it is not complete: the subset of negative numbers doesn't have a supremum within that set; likewise, the set of positive numbers doesn't have an infimum.
- Bounded and complete: just take any powerset lattice.
- Neither bounded nor complete: take the natural numbers with the usual order.
- Not bounded but complete: there is no such thing. If a lattice $L$ is complete, then it is bounded by $0_L = bigwedge L$ and $1_L = bigvee L$.
answered Dec 26 '18 at 10:58
amrsaamrsa
3,7852618
answered Dec 26 '18 at 10:58
amrsaamrsa
3,7852618
answered Dec 26 '18 at 10:58
amrsaamrsa
3,7852618
answered Dec 26 '18 at 10:58
amrsaamrsa
3,7852618
3,7852618
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