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.
22.
It consists of a partially ordered set in which every two elements have a unique supremum ( also called a least upper bound or meet ).
23.
Present a unified framework for pre-processing Directed Acyclic Graphs ( DAGs ) to compute the least upper bound of two vertices in constant time.
24.
The least upper bound principle has also been analyzed in the program of reverse mathematics, where the exact strength of this principle has been determined.
25.
In order theory, this property can be generalized to a notion of dense and has the least upper bound property is called a linear continuum.
26.
In the axiomatic theory of real numbers, the non-existence of nonzero infinitesimal real numbers is implied by the least upper bound property as follows.
27.
If every subset that has " some " upper bounds has also a least upper bound, then the respective poset is called bounded complete.
28.
In general, subsets of this order do not have least upper bounds : just consider the set { 0.1, 0.3 }.
29.
The only real number axiom that does not follow easily from the definitions is the completeness of d ", i . e . the least upper bound property.
30.
The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers : the existence of least upper bounds or " suprema ".
