bounded lattice
A lattice is said to be if there is an element such that for all . Dually, is if there exists an element such that for all . A bounded lattice is one that is 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 and as described above.
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and for all .
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and for all .
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As a result, and , if they exist, are necessarily unique. For if there is another such a pair and , then . Similarly .
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is called the bottom of and is called the top of .
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is a lattice interval and can be written as .
Remark. More generally, a poset is said to be bounded if it has both a greatest element and a least element .
Title | bounded lattice |
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Canonical name | BoundedLattice |
Date of creation | 2013-03-22 15:02:28 |
Last modified on | 2013-03-22 15:02:28 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 13 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 06B05 |
Classification | msc 06A06 |
Defines | top |
Defines | bottom |
Defines | bounded poset |