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bounded lattice
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(Definition)
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A lattice  is said to be bounded from below if there is an element  such that  for all  . Dually,  is bounded from above if there exists an element  such that  for all  . A bounded lattice is one that is bounded both from above and below.
For example, any finite lattice  is bounded, as  and
 , being join and meet of finitely many elements, exist.
 and
 .
Remarks. Let  be a bounded lattice with 0 and  as described above.
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 and  for all  .
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 and  for all  .
- As a result, 0 and
 , if they exist, are necessarily unique. For if there is another such a pair
 and
 , then
 . Similarly
 .
- 0 is called the bottom of
 and  is called the top of  .
 is a lattice interval and can be written as ![$ [0,1]$](http://images.planetmath.org:8080/cache/objects/6755/l2h/img31.png "$ [0,1]$") .
Remark. More generally, a poset  is said to be bounded if it has both a greatest element  and a least element 0.
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"bounded lattice" is owned by CWoo.
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(view preamble)
| Also defines: |
top, bottom, bounded poset |
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Cross-references: least element, greatest element, poset, lattice interval, meet, join, bounded, finite, lattice
There are 42 references to this entry.
This is version 10 of bounded lattice, born on 2005-02-16, modified 2007-04-29.
Object id is 6755, canonical name is BoundedLattice.
Accessed 4609 times total.
Classification:
| AMS MSC: | 06B05 (Order, lattices, ordered algebraic structures :: Lattices :: Structure theory) | | | 06A06 (Order, lattices, ordered algebraic structures :: Ordered sets :: Partial order, general) |
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Pending Errata and Addenda
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