Order Theory - Subsets of Ordered Sets

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 yx}. A set that is equal to its upper closure is called an upper set. Lower sets are defined dually.

More complicated lower subsets are ideals, which have the additional property that each two of their elements have an upper bound within the ideal. Their duals are given by filters. A related concept is that of a 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 chain. The opposite notion, the antichain, is a subset that contains no two comparable elements; i.e. that is a discrete order.

Read more about this topic:  Order Theory

Famous quotes containing the words ordered and/or sets:

    Then he rang the bell and ordered a ham sandwich. When the maid placed the plate on the table, he deliberately looked away but as soon as the door had shut, he grabbed the sandwich with both hands, immediately soiled his fingers and chin with the hanging margin of fat and, grunting greedily, began to much.
    Vladimir Nabokov (1899–1977)

    The vain man does not wish so much to be prominent as to feel himself prominent; he therefore disdains none of the expedients for self-deception and self-outwitting. It is not the opinion of others that he sets his heart on, but his opinion of their opinion.
    Friedrich Nietzsche (1844–1900)