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.
|Date of creation||2013-03-22 17:01:08|
|Last modified on||2013-03-22 17:01:08|
|Last modified by||CWoo (3771)|