It also has least upper bounds of chains.

It may help to point out that the increasing sequences produced by sequential programs all have least upper bounds.

These are functions that preserve the order structure, and that preserve least upper bounds.

Limits and continuity; least upper bounds, intermediate and extreme value theorems.

Theory curbing selects such least upper bounds models in addition to the ones selected by circumscription.

Note that while the ordering on is given by the subset relation, least upper bounds do not in general coincide with unions.

As in the case of limits of sequences, least upper bounds of directed sets do not always exist.

The fact that there exist increasing sequences without least upper bounds may seem strange to those accustomed to thinking about the semantics of sequential programs.

More generally, we would like to restrict to a certain subset of elements as being sufficient for getting all other elements as least upper bounds.

Least upper bounds (suprema, ) and greatest lower bounds (infima, )

In this case the products and coproducts correspond to greatest lower bounds (meets) and least upper bounds (joins).

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of least upper bounds or suprema.

From the article on Power domains: is the collection of downward-closed subsets of domain that are also closed under existing least upper bounds of directed sets in .

This inclusion is done until the set of models is closed, in the sense that it includes all least upper bounds of all sets of models it contains.

Thus, it applies to chain complete posets, but is more general in that it allows chains that have upper bounds but do not have least upper bounds.

The construction of the real numbers using Dedekind cuts takes advantage of this failure by defining the irrational numbers as the least upper bounds of certain subsets of the rationals.

The motivation for considering completeness properties derives from the great importance of suprema (least upper bounds, joins, "") and infima (greatest lower bounds, meets, "") to the theory of partial orders.

A continuous semiring is similarly defined as one for which the addition monoid is a continuous monoid: that is, partially ordered with the least upper bounds property, and for which addition and multiplication respect order and suprema.