(Last modified Thu Apr 24 10:25 2008)
Let P be an ordered set. P is a cpo iff for every pair x,y∈P, their LUB x∨y is an elements of P.
Another way of saying the same thing: an ordered set P is a cpo if it is closed under the cup operation.
A cpo with a least element 
is called a cpo with bottom.
Some authors use "cpo" to mean "cpo with bottom".
A cpo is sort of a "half-lattice" (see below). A lattice has LUBs and GLBs, a cpo has LUBs only.
Examples:
Every powerset contains 
;
⊆S for every subset S∈P, so is a lower bound;
and no other set is a lower bound for P, so is the GLB
of the entire powerset P, and thus is .
See Figure 1.
Let P be an ordered set. P is a lattice iff for every pair x,y∈P, their LUB x∨y and GLB x∧y are both elements of P.
Another way of saying the same thing: an ordered set P is a lattice if it is closed under the cap and cup operations.
A lattice may have a top and/or a bottom, but an infinite lattice need not have either. A finite lattice always has a top and a bottom.
Examples:
The LUB
(GLB) of two sets X and Y is
their union
(intersection),
and a powerset is closed
under intersection (union),
so P is a lattice.
P is finite and thus has a top and a bottom;
is the set S of which P is the powerset,
and
is the empty set .
See ordered set Figure 2.
The LUB (GLB) of two integers x and y is the lesser (greater) of the two, and so is an integer; thus the integers are closed under LUB (GLB). The integers are an infinite set and so need not have a top or a bottom, and in fact do not; there is no greatest or least integer. See ordered set Figure 1.
B. A. Davey and H. A. Priestley. Introduction to Lattices and Order. Cambridge University Press, 2002.
