A lattice is any 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. In some literature, is required to be non-empty.
A sublattice of is a subposet of which is a lattice, that is, which is closed under the operations and as defined in .
Once 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 (http://planetmath.org/DistributiveLattice). While many nice lattices, such as face lattices of polytopes, are distributive, there are also important classes of lattices, such as partition lattices (http://planetmath.org/PartitionLattice), that are usually not distributive.
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
Remark. Alternatively, a lattice can be defined as an algebraic system. Please see the link below for details.