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special elements in a lattice
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(Definition)
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Let  be a lattice and  is said to be
for all  . There are also dual notions of the three types mentioned above, simply by exchanging  and  in the definitions. So a dually distributive element  is one where
 for all  , and a dually standard element is similarly defined. However, a dually neutral element is the same as a neutral element.
Remarks For any  , suppose  is the property in  such that  iff
 and
 imply  for all  .
- A standard element is distributive. Conversely, a distributive satisfying
 is standard.
- A neutral element is distributive (and consequently dually distributive). Conversely, a distributive and dually distributive element that satisfies
 is neutral.
- 1
- G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).
- 2
- G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).
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"special elements in a lattice" is owned by CWoo.
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(view preamble | get metadata)
| Also defines: |
distributive element, standard element, neutral element, dually distributive, dually standard |
| Keywords: |
distributive, standard, neutral |
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Cross-references: imply, iff, property, definitions, types, distributive, lattice
There is 1 reference to this entry.
This is version 3 of special elements in a lattice, born on 2007-02-17, modified 2007-04-21.
Object id is 8923, canonical name is SpecialElementsInALattice.
Accessed 2042 times total.
Classification:
| AMS MSC: | 06B99 (Order, lattices, ordered algebraic structures :: Lattices :: Miscellaneous) |
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Pending Errata and Addenda
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