Editing Order theory (section)

== Subsets of ordered sets ==
In an ordered set, one can define many types of special subsets based on the given order. A simple example are '''upper sets'''; i.e. sets that contain all elements that are above them in the order. Formally, the '''upper closure''' of a set ''S'' in a poset ''P'' is given by the set {''x'' in ''P'' | there is some ''y'' in ''S'' with ''y'' ≤ ''x''}. A set that is equal to its upper closure is called an upper set. '''Lower sets''' are defined dually.

More complicated lower subsets are [[ideal (order theory)|ideals]], which have the additional property that each two of their elements have an upper bound within the ideal. Their duals are given by [[filter (mathematics)|filters]]. A related concept is that of a [[directed set|directed subset]], which like an ideal contains upper bounds of finite subsets, but does not have to be a lower set. Furthermore, it is often generalized to preordered sets.

A subset which is – as a sub-poset – linearly ordered, is called a [[total order#Chains|chain]]. The opposite notion, the [[antichain]], is a subset that contains no two comparable elements; i.e. that is a discrete order.