Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Upper set
Summary
In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X) of a partially ordered set is a subset with the following property: if s is in S and if x in X is larger than s (that is, if ), then x is in S. In other words, this means that any x element of X that is to some element of S is necessarily also an element of S.
The term lower set (also called a downward closed set, down set, decreasing set, initial segment, or semi-ideal) is defined similarly as being a subset S of X with the property that any element x of X that is to some element of S is necessarily also an element of S.
Let be a preordered set.
An in (also called an , an , or an set) is a subset that is "closed under going up", in the sense that
for all and all if then
The dual notion is a (also called a , , , , or ), which is a subset that is "closed under going down", in the sense that
for all and all if then
The terms or are sometimes used as synonyms for lower set. This choice of terminology fails to reflect the notion of an ideal of a lattice because a lower set of a lattice is not necessarily a sublattice.
Every partially ordered set is an upper set of itself.
The intersection and the union of any family of upper sets is again an upper set.
The complement of any upper set is a lower set, and vice versa.
Given a partially ordered set the family of upper sets of ordered with the inclusion relation is a complete lattice, the upper set lattice.
Given an arbitrary subset of a partially ordered set the smallest upper set containing is denoted using an up arrow as (see upper closure and lower closure).
Dually, the smallest lower set containing is denoted using a down arrow as
A lower set is called principal if it is of the form where is an element of
Every lower set of a finite partially ordered set is equal to the smallest lower set containing all maximal elements of
where denotes the set containing the maximal elements of
A directed lower set is called an order ideal.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.