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A lattice is any non-empty poset  in which any two elements  and  have a least upper bound,  , and a greatest lower bound,  . The operation  is called meet, and the operation  is called join. A sublattice of  is a subposet of  which is a lattice, that is, which is closed under the operations  and  as defined in  .
The operations of meet and join are idempotent, commutative, associative, and absorptive:
 and   if and only if   is defined, it is not hard to see that  iff  as well (one direction goes like:
 , while the other direction is the dual of the first).
Conspicuously absent from the above list of properties is distributivity. While many nice lattices, such as face lattices of polytopes, are distributive, there are also important classes of lattices, such as partition lattices, that are usually not distributive.
There is also alternative definition of a lattice (where lattice is defined as an algebraic structure with operations  and  ).
Lattices, like posets, can be visualized by diagrams called Hasse diagrams. Below are two diagrams, both posets. The one on the left is a lattice, while the one on the right is not:
The vertices of a lattice diagram can also be labelled, so the lattice diagram looks like
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"lattice" is owned by mps. [ full author list (4) | owner history (1) ]
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(view preamble)
Cross-references: vertices, right, Hasse diagrams, algebraic structure, polytopes, face, iff, relation, partial order, band, associative, commutative, idempotent, closed under, join, meet, operation, greatest lower bound, least upper bound, poset
There are 115 references to this entry.
This is version 17 of lattice, born on 2002-02-24, modified 2007-12-04.
Object id is 2593, canonical name is Lattice.
Accessed 13166 times total.
Classification:
| AMS MSC: | 06B99 (Order, lattices, ordered algebraic structures :: Lattices :: Miscellaneous) | | | 03G10 (Mathematical logic and foundations :: Algebraic logic :: Lattices and related structures) |
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Pending Errata and Addenda
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