A lattice is a partical ordered

set (set with a

partial order), where every

subset containing 2

elements (subsets {a,b}) has a

least upper bound (supremum, sup) and a

greatest lower bound (infimum, inf).

A lower bound i is a infimum of a set S,
iff for every x with for all y of S x <= y , x <= i holds.
Dito for supremum with >=.

A lattice is called complete iff every subset has a supremum and an infimum.

A set whose all subsets {a,b} have a supremum is called upper half lattice. In a lower half lattice only the infimums must exist. Upper/lower half lattices are called complete iff the supremums/infimums of all subsets exist.

Examples: Set of subsets of a set is a complete lattice.

Boolean algebras are complete lattices (in fact lattices can be used, along with other constraints, to define a boolean algebra)

The real numbers are a lattice, but not complete.