bounded complete

Let P be a poset. Recall that a subset S of P is called bounded from above if there is an element aP such that, for every sS, sa.

A poset P is said to be bounded complete if every subset which is bounded from above has a supremumMathworldPlanetmathPlanetmath.

Remark. Since it is not required that the subset be non-empty, we see that P has a bottom. This is because the empty setMathworldPlanetmath is vacuously bounded from above, and therefore has a supremum. However, this supremum is less than or equal to every member of P, and hence it is the least element of P.

Clearly, any complete latticeMathworldPlanetmath is bounded complete. An example of a non-complete bounded complete poset is any closed subset of of the form [a,), where a. In additionPlanetmathPlanetmath, 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 P 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.

Title bounded complete
Canonical name BoundedComplete
Date of creation 2013-03-22 17:01:08
Last modified on 2013-03-22 17:01:08
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 06A12
Classification msc 06B23
Classification msc 03G10
Related topic CompletenessPrinciple
Defines Dedekind complete