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Are the real numbers a lattice?

2

$begingroup$

A lattice is a set with a partial order, where every pair has a unique upper and lower bound.

As far as I can tell, there is nothing in the definition that forces the set to be discrete. In particular, the real numbers with their usual partial order and upper/lower bounds seems to fit the definition. But all the examples I've come across are discrete, which makes me wonder if I've missed something. (e.g. Wikipedia)

lattice-orders

share|cite|improve this question

asked Apr 8 at 20:20

prdnrprdnr

956

$endgroup$

• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  You're not missing anything. Every totally ordered set is a lattice.
  $endgroup$
  – Wojowu
  Apr 8 at 20:23
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You didn't miss anything, the real numbers form a lattice.
  $endgroup$
  – Javi
  Apr 8 at 20:24
  

  
  
  
  

add a comment | 

2

$begingroup$

A lattice is a set with a partial order, where every pair has a unique upper and lower bound.

As far as I can tell, there is nothing in the definition that forces the set to be discrete. In particular, the real numbers with their usual partial order and upper/lower bounds seems to fit the definition. But all the examples I've come across are discrete, which makes me wonder if I've missed something. (e.g. Wikipedia)

lattice-orders

share|cite|improve this question

asked Apr 8 at 20:20

prdnrprdnr

956

$endgroup$

• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  You're not missing anything. Every totally ordered set is a lattice.
  $endgroup$
  – Wojowu
  Apr 8 at 20:23
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You didn't miss anything, the real numbers form a lattice.
  $endgroup$
  – Javi
  Apr 8 at 20:24
  

  
  
  
  

add a comment | 

$begingroup$

A lattice is a set with a partial order, where every pair has a unique upper and lower bound.

As far as I can tell, there is nothing in the definition that forces the set to be discrete. In particular, the real numbers with their usual partial order and upper/lower bounds seems to fit the definition. But all the examples I've come across are discrete, which makes me wonder if I've missed something. (e.g. Wikipedia)

lattice-orders

share|cite|improve this question

asked Apr 8 at 20:20

prdnrprdnr

956

$endgroup$

share|cite|improve this question

asked Apr 8 at 20:20

prdnrprdnr

956

share|cite|improve this question

asked Apr 8 at 20:20

prdnrprdnr

956

asked Apr 8 at 20:20

prdnrprdnr

956

asked Apr 8 at 20:20

prdnrprdnr

956

1 Answer
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3

$begingroup$

In a lattice, every pair must have a least upper bound (called join) and greatest lower bound (called meet), that is different from it being unique. There can be many upper bounds or lower bounds. The join or meet of a pair however, is unique.

The reals are in fact a particularly nice kind of lattice:

• They are totally ordered, which means that any two elements $a, b in mathbbR$ can be compared and therefore the join and meet can be computed very easily by using $max(a, b)$ and $min(a, b)$ respectively.


• When we take a closed interval (e.g. $[0,1]$), then that interval is complete. This means that every subset $A subseteq [0,1]$ has a least upper bound, in this case given by its supremum $sup A$. Similarly for a greatest lower bound and the infimum.

share|cite|improve this answer

answered Apr 8 at 20:33

Mark KamsmaMark Kamsma

3617

New contributor

Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

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3

$begingroup$

In a lattice, every pair must have a least upper bound (called join) and greatest lower bound (called meet), that is different from it being unique. There can be many upper bounds or lower bounds. The join or meet of a pair however, is unique.

The reals are in fact a particularly nice kind of lattice:

• They are totally ordered, which means that any two elements $a, b in mathbbR$ can be compared and therefore the join and meet can be computed very easily by using $max(a, b)$ and $min(a, b)$ respectively.


• When we take a closed interval (e.g. $[0,1]$), then that interval is complete. This means that every subset $A subseteq [0,1]$ has a least upper bound, in this case given by its supremum $sup A$. Similarly for a greatest lower bound and the infimum.

share|cite|improve this answer

answered Apr 8 at 20:33

Mark KamsmaMark Kamsma

3617

New contributor

Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

3

$begingroup$

In a lattice, every pair must have a least upper bound (called join) and greatest lower bound (called meet), that is different from it being unique. There can be many upper bounds or lower bounds. The join or meet of a pair however, is unique.

The reals are in fact a particularly nice kind of lattice:

• They are totally ordered, which means that any two elements $a, b in mathbbR$ can be compared and therefore the join and meet can be computed very easily by using $max(a, b)$ and $min(a, b)$ respectively.


• When we take a closed interval (e.g. $[0,1]$), then that interval is complete. This means that every subset $A subseteq [0,1]$ has a least upper bound, in this case given by its supremum $sup A$. Similarly for a greatest lower bound and the infimum.

share|cite|improve this answer

answered Apr 8 at 20:33

Mark KamsmaMark Kamsma

3617

New contributor

Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

$begingroup$

In a lattice, every pair must have a least upper bound (called join) and greatest lower bound (called meet), that is different from it being unique. There can be many upper bounds or lower bounds. The join or meet of a pair however, is unique.

The reals are in fact a particularly nice kind of lattice:

• They are totally ordered, which means that any two elements $a, b in mathbbR$ can be compared and therefore the join and meet can be computed very easily by using $max(a, b)$ and $min(a, b)$ respectively.


• When we take a closed interval (e.g. $[0,1]$), then that interval is complete. This means that every subset $A subseteq [0,1]$ has a least upper bound, in this case given by its supremum $sup A$. Similarly for a greatest lower bound and the infimum.

share|cite|improve this answer

answered Apr 8 at 20:33

Mark KamsmaMark Kamsma

3617

New contributor

Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this answer

answered Apr 8 at 20:33

Mark KamsmaMark Kamsma

3617

New contributor

Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

answered Apr 8 at 20:33

Mark KamsmaMark Kamsma

3617

New contributor

Mark Kamsma is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

answered Apr 8 at 20:33

Mark KamsmaMark Kamsma

3617