lattice

A lattice is any poset $L$ in which any two elements $x$ and $y$ have a least upper bound, $x\vee y$, and a greatest lower bound, $x\wedge y$. The operation $\wedge$ is called meet, and the operation $\vee$ is called join. In some literature, $L$ is required to be non-empty.

A sublattice of $L$ is a subposet of $L$ which is a lattice, that is, which is closed under the operations $\wedge$ and $\vee$ as defined in $L$.

The operations of meet and join are idempotent, commutative, associative, and absorptive:

$$x\wedge (y\vee x)=x\text{ and }x\vee (y\wedge x)=x.$$

Thus a lattice is a commutative band with either operation. The partial order relation can be recovered from meet and join bydefining

$$x\le y\text{ if and only if }x\wedge y=x.$$

Once $\le$ is defined, it is not hard to see that $x\le y$ iff $x\vee y=y$ as well (one direction goes like: $x\vee y=(x\wedge y)\vee y=y\vee (x\wedge y)=y\vee (y\wedge x)=y$, 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.