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Bounded Complete
Let 
be a poset. Recall that a subset
of 
is called bounded from above if there is an element 
such that, for every 
, 
.
A poset 
is said to be bounded complete if every subset which is bounded from above has a supremum.
Remark. Since it is not required that the subset be non-empty, we see that 
has a bottom. This is because the empty set
is vacuously
bounded from above, and therefore has a supremum. However, this supremum is less than or equal to every member of 
, and hence it is the least element
of 
.
Clearly, any complete lattice
is bounded complete. An example of a non-complete bounded complete poset is any closed subset
of
of the form
, where
. In addition, arbitrary products
of bounded complete posets is also bounded complete.
It can be shown that a poset is a bounded complete dcpo iff it is a complete semilattice.
Remark. A weaker concept is that of Dedekind completeness: A poset 
is Dedekind complete if every non-empty subset bounded from above has a supremum. An obvious
example is
, which is Dedekind complete but not bounded complete (as it has no bottom). Dedekind completeness is more commonly known as the least upper bound property.