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Lattice
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.
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:
![$\displaystyle \entrymodifiers={[o]} \xymatrix @!=1pt {
& & \bullet \ar@{-}[ld] ...
...[ld] \\
& \bullet \ar@{-}[rd] & & \bullet \ar@{-}[ld] & \\
& & \bullet & & }
$](http://myyn.org/static/assets/mathbrowser/article/531/images/img21.png)
![$\displaystyle \xymatrix @!=1pt {
& & a \ar@{-}[ld] \ar@{-}[rd] & & \\
& b \ar@...
...[rd] & & f \ar@{-}[ld] \\
& g \ar@{-}[rd] & & h \ar@{-}[ld] & \\
& & i & & }
$](http://myyn.org/static/assets/mathbrowser/article/531/images/img22.png)
Remark. Alternatively, a lattice can be defined as an algebraic system. Please see the link below for details.