Editing Bounded complete poset

In the [[mathematics|mathematical]] field of [[order theory]], a [[partially ordered set]] is '''bounded complete''' if all of its [[subset]]s that have some [[upper bound]] also have a [[least upper bound]]. Such a partial order can also be called '''consistently''' or '''coherently complete''' ([[#Visser2004|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]]. 

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 (order theory)|completeness]] properties, which are detailed in the article on [[completeness (order theory)|completeness in order theory]]. The term ''bounded poset'' is sometimes used to refer to a partially ordered set that has both a [[greatest element and least element|least element and greatest element]]. Hence it is important to distinguish between a bounded-complete poset and a bounded [[complete partial order]] (cpo).

For a typical example of a bounded-complete poset, consider the set of all finite [[decimal number]]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]] of words: a decimal number ''n'' is below some other number ''m'' if there is some [[string (computer science)|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]]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]]s, which provide many other examples for bounded-complete posets.

==References==
* <cite id=Visser2004>Visser, A. (2004) 'Semantics and the Liar Paradox' in: [[Dov Gabbay|D.M. Gabbay]] and F. Günther (ed.) Handbook of Philosophical Logic, 2nd Edition, Volume 11, pp.&nbsp;149 – 240</cite>

[[Category:Order theory]]