lattice

A lattice is any poset $L$ in which any two elements $x$ and $y$ have a least upper bound, $x\lor y$ , and a greatest lower bound, $x\land y$ . The operation $\land$ is called meet, and the operation $\lor$ 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 $\land$ and $\lor$ as defined in $L$ .

The operations of meet and join are idempotent, commutative, associative, and absorptive:$$x\land (y\lor x)=x\mbox{ and }x\lor (y\land x)=x.$$ Thus a lattice is a commutative band with either operation. The partial order relation can be recovered from meet and join by defining$$x\le y \text{\ if and only if\ } x\land y = x$$ Once $\le$ is defined, it is not hard to see that $x\le y$ iff $x\lor y=y$ as well (one direction goes like: $x\lor y= (x\land y)\lor y=y\lor (x\land y)=y\lor (y\land x)=y$ , 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.

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.