A latticeMathworldPlanetmathPlanetmath is any poset L in which any two elements x and y have a least upper bound, xy, and a greatest lower boundMathworldPlanetmath, xy. The operationMathworldPlanetmath is called meet, and the operation 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 and as defined in L.

The operations of meet and join are idempotentPlanetmathPlanetmath, commutativePlanetmathPlanetmathPlanetmath, associative, and absorptive:

x(yx)=x and x(yx)=x.

Thus a lattice is a commutative band with either operation. The partial orderMathworldPlanetmath relationMathworldPlanetmathPlanetmath can be recovered from meet and join by defining

xy if and only if xy=x.

Once is defined, it is not hard to see that xy iff xy=y as well (one direction goes like: xy=(xy)y=y(xy)=y(yx)=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 partitionPlanetmathPlanetmath 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:

\entrymodifiers=[o]\xymatrix@!=1pt&&\ar@-[ld]\ar@-[rd]&&&\ar@-[ld]\ar@-[rd]&&\ar@-[ld]\ar@-[rd]&\ar@-[rd]&&\ar@-[ld]\ar@-[rd]&&\ar@-[ld]&\ar@-[rd]&&\ar@-[ld]&&&&&       \entrymodifiers=[o]\xymatrix@!=1pt&&\ar@-[ld]\ar@-[rd]&&&\ar@-[ld]\ar@-[rd]\ar@-[d]&&\ar@-[ld]\ar@-[rd]\ar@-[d]&\ar@-[rd]&\ar@-[d]&\ar@-[ld]\ar@-[rd]&\ar@-[d]&\ar@-[ld]&\ar@-[rd]&&\ar@-[ld]&&&&&

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.

Title lattice