Example of a uniquely complemented lattice?
$begingroup$
I'm approaching lattices. I have understood the definition of a Lattice, a Complete Lattice and a Bounded Lattice.
Theoretically, even the definition of a complemented lattice doesn't seem difficult, but I couldn't get any satisfying examples.
In fact, every time I searched for it, I only got pictures of graphs with letters on vertices and a $0$ and a $1$ on the remaining vertices.
I'd love to have a number set example if it's possible, or a non-strictly mathematical (real-life set) example, or just something that is really intuitive.
order-theory lattice-orders
edited Dec 23 '18 at 20:25
the_fox
2,90031538
asked Dec 18 '18 at 11:09
Gabriele ScarlattiGabriele Scarlatti
370212
$endgroup$
add a comment |
$begingroup$
I'm approaching lattices. I have understood the definition of a Lattice, a Complete Lattice and a Bounded Lattice.
Theoretically, even the definition of a complemented lattice doesn't seem difficult, but I couldn't get any satisfying examples.
In fact, every time I searched for it, I only got pictures of graphs with letters on vertices and a $0$ and a $1$ on the remaining vertices.
I'd love to have a number set example if it's possible, or a non-strictly mathematical (real-life set) example, or just something that is really intuitive.
order-theory lattice-orders
edited Dec 23 '18 at 20:25
the_fox
2,90031538
asked Dec 18 '18 at 11:09
Gabriele ScarlattiGabriele Scarlatti
370212
$endgroup$
add a comment |
$begingroup$
I'm approaching lattices. I have understood the definition of a Lattice, a Complete Lattice and a Bounded Lattice.
Theoretically, even the definition of a complemented lattice doesn't seem difficult, but I couldn't get any satisfying examples.
In fact, every time I searched for it, I only got pictures of graphs with letters on vertices and a $0$ and a $1$ on the remaining vertices.
I'd love to have a number set example if it's possible, or a non-strictly mathematical (real-life set) example, or just something that is really intuitive.
order-theory lattice-orders
edited Dec 23 '18 at 20:25
the_fox
2,90031538
asked Dec 18 '18 at 11:09
Gabriele ScarlattiGabriele Scarlatti
370212
$endgroup$
I'm approaching lattices. I have understood the definition of a Lattice, a Complete Lattice and a Bounded Lattice.
Theoretically, even the definition of a complemented lattice doesn't seem difficult, but I couldn't get any satisfying examples.
In fact, every time I searched for it, I only got pictures of graphs with letters on vertices and a $0$ and a $1$ on the remaining vertices.
I'd love to have a number set example if it's possible, or a non-strictly mathematical (real-life set) example, or just something that is really intuitive.
order-theory lattice-orders
order-theory lattice-orders
edited Dec 23 '18 at 20:25
the_fox
2,90031538
asked Dec 18 '18 at 11:09
Gabriele ScarlattiGabriele Scarlatti
370212
edited Dec 23 '18 at 20:25
the_fox
2,90031538
asked Dec 18 '18 at 11:09
Gabriele ScarlattiGabriele Scarlatti
370212
edited Dec 23 '18 at 20:25
the_fox
2,90031538
edited Dec 23 '18 at 20:25
the_fox
2,90031538
edited Dec 23 '18 at 20:25
the_fox
2,90031538
2,90031538
asked Dec 18 '18 at 11:09
Gabriele ScarlattiGabriele Scarlatti
370212
asked Dec 18 '18 at 11:09
Gabriele ScarlattiGabriele Scarlatti
370212
asked Dec 18 '18 at 11:09
Gabriele ScarlattiGabriele Scarlatti
370212
370212
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
If you're interested in uniquely complemented lattices, then
- In the distributive case (all the distributive complemented lattices), they're Boolean, that is, they are the lattice reducts of Boolean algebras (so, in the finite case, which might be what you're interested in, they have the shape $mathbf 2^n$, for some natural number $n$);
- In the non-distributive case, these seem very elusive creatures, although they exist (but they are infinite).
See the answer to this question with links to related papers.
To emphasize how elusive these seem to be, note this sentence in the linked paper by Grätzer (end of part 2):
It is interesting how little we know about a subject on which we have published so many papers.
Ask any question and probably we do not know the answer.
answered Dec 23 '18 at 18:31
amrsaamrsa
3,7852618
$endgroup$
add a comment |
$begingroup$
According to this Wolfram entry, any Boolean algebra is a uniquely complemented lattice.
http://mathworld.wolfram.com/UniquelyComplementedLattice.html
answered Dec 18 '18 at 11:16
coffeemathcoffeemath
2,8971415
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add a comment |
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
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active
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$begingroup$
If you're interested in uniquely complemented lattices, then
- In the distributive case (all the distributive complemented lattices), they're Boolean, that is, they are the lattice reducts of Boolean algebras (so, in the finite case, which might be what you're interested in, they have the shape $mathbf 2^n$, for some natural number $n$);
- In the non-distributive case, these seem very elusive creatures, although they exist (but they are infinite).
See the answer to this question with links to related papers.
To emphasize how elusive these seem to be, note this sentence in the linked paper by Grätzer (end of part 2):
It is interesting how little we know about a subject on which we have published so many papers.
Ask any question and probably we do not know the answer.
answered Dec 23 '18 at 18:31
amrsaamrsa
3,7852618
$endgroup$
add a comment |
$begingroup$
If you're interested in uniquely complemented lattices, then
- In the distributive case (all the distributive complemented lattices), they're Boolean, that is, they are the lattice reducts of Boolean algebras (so, in the finite case, which might be what you're interested in, they have the shape $mathbf 2^n$, for some natural number $n$);
- In the non-distributive case, these seem very elusive creatures, although they exist (but they are infinite).
See the answer to this question with links to related papers.
To emphasize how elusive these seem to be, note this sentence in the linked paper by Grätzer (end of part 2):
It is interesting how little we know about a subject on which we have published so many papers.
Ask any question and probably we do not know the answer.
answered Dec 23 '18 at 18:31
amrsaamrsa
3,7852618
$endgroup$
add a comment |
$begingroup$
If you're interested in uniquely complemented lattices, then
- In the distributive case (all the distributive complemented lattices), they're Boolean, that is, they are the lattice reducts of Boolean algebras (so, in the finite case, which might be what you're interested in, they have the shape $mathbf 2^n$, for some natural number $n$);
- In the non-distributive case, these seem very elusive creatures, although they exist (but they are infinite).
See the answer to this question with links to related papers.
To emphasize how elusive these seem to be, note this sentence in the linked paper by Grätzer (end of part 2):
It is interesting how little we know about a subject on which we have published so many papers.
Ask any question and probably we do not know the answer.
answered Dec 23 '18 at 18:31
amrsaamrsa
3,7852618
$endgroup$
If you're interested in uniquely complemented lattices, then
- In the distributive case (all the distributive complemented lattices), they're Boolean, that is, they are the lattice reducts of Boolean algebras (so, in the finite case, which might be what you're interested in, they have the shape $mathbf 2^n$, for some natural number $n$);
- In the non-distributive case, these seem very elusive creatures, although they exist (but they are infinite).
See the answer to this question with links to related papers.
To emphasize how elusive these seem to be, note this sentence in the linked paper by Grätzer (end of part 2):
It is interesting how little we know about a subject on which we have published so many papers.
Ask any question and probably we do not know the answer.
answered Dec 23 '18 at 18:31
amrsaamrsa
3,7852618
answered Dec 23 '18 at 18:31
amrsaamrsa
3,7852618
answered Dec 23 '18 at 18:31
amrsaamrsa
3,7852618
answered Dec 23 '18 at 18:31
amrsaamrsa
3,7852618
3,7852618
add a comment |
add a comment |
$begingroup$
According to this Wolfram entry, any Boolean algebra is a uniquely complemented lattice.
http://mathworld.wolfram.com/UniquelyComplementedLattice.html
answered Dec 18 '18 at 11:16
coffeemathcoffeemath
2,8971415
$endgroup$
add a comment |
$begingroup$
According to this Wolfram entry, any Boolean algebra is a uniquely complemented lattice.
http://mathworld.wolfram.com/UniquelyComplementedLattice.html
answered Dec 18 '18 at 11:16
coffeemathcoffeemath
2,8971415
$endgroup$
add a comment |
$begingroup$
According to this Wolfram entry, any Boolean algebra is a uniquely complemented lattice.
http://mathworld.wolfram.com/UniquelyComplementedLattice.html
answered Dec 18 '18 at 11:16
coffeemathcoffeemath
2,8971415
$endgroup$
According to this Wolfram entry, any Boolean algebra is a uniquely complemented lattice.
http://mathworld.wolfram.com/UniquelyComplementedLattice.html
answered Dec 18 '18 at 11:16
coffeemathcoffeemath
2,8971415
answered Dec 18 '18 at 11:16
coffeemathcoffeemath
2,8971415
answered Dec 18 '18 at 11:16
coffeemathcoffeemath
2,8971415
answered Dec 18 '18 at 11:16
coffeemathcoffeemath
2,8971415
2,8971415
add a comment |
add a comment |
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