In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...

, a

partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...

is bounded complete if all of its

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

s that have some

upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an eleme ...

also have a

least upper bound In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...

. Such a partial order can also be called consistently or coherently complete ( Visser 2004, p. 182), since any upper bound of a set can be interpreted as some consistent (non-contradictory) piece of information that extends all the information present in the set. Hence the presence of some upper bound in a way guarantees the consistency of a set. Bounded completeness then yields the existence of a least upper bound of any "consistent" subset, which can be regarded as the most general piece of information that captures all the knowledge present within this subset. This view closely relates to the idea of information ordering that one typically finds in

domain theory Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer ...

. Formally, a partially ordered set (''P'', ≤) is ''bounded complete'' if the following holds for any subset ''S'' of ''P'' : : If ''S'' has some upper bound, then it also has a least upper bound. Bounded completeness has various relationships to other completeness properties, which are detailed in the article on completeness in order theory. The term ''bounded poset'' is sometimes used to refer to a partially ordered set that has both a least element and greatest element. Hence it is important to distinguish between a bounded-complete poset and a bounded

complete partial order In mathematics, the phrase complete partial order is variously used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties. Complete partial orders play a central rol ...

(cpo). For a typical example of a bounded-complete poset, consider the set of all finite

decimal number The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...

s starting with "0." (like 0.1, 0.234, 0.122) together with all infinite such numbers (like the decimal representation 0.1111... of 1/9). Now these elements can be ordered based on the

prefix order In mathematics, especially order theory, a prefix ordered set generalizes the intuitive concept of a tree by introducing the possibility of continuous progress and continuous branching. Natural prefix orders often occur when considering dynamical sy ...

of words: a decimal number ''n'' is below some other number ''m'' if there is some string of digits w such that ''n''w = ''m''. For example, 0.2 is below 0.234, since one can obtain the latter by appending the string "34" to 0.2. The infinite decimal numbers are the

maximal element In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is defin ...

s within this order. In general, subsets of this order do not have least upper bounds: just consider the set {0.1, 0.3}. Looking back at the above intuition, one might say that it is not consistent to assume that some number starts both with 0.1 and with 0.3. However, the order is still bounded complete. In fact, it is even an example of a more specialized class of structures, the

Scott domain In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded-complete cpo. They are named in honour of Dana S. Scott, who was the first to study these structures at the advent of domain theory. Scott domains a ...

s, which provide many other examples for bounded-complete posets.

References

* Visser, A. (2004) ‘Semantics and the Liar Paradox’ in: D.M. Gabbay and F. Günther (ed.) Handbook of Philosophical Logic, 2nd Edition, Volume 11, pp. 149 – 240 Order theory