In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an upper set (also called an upward closed set or just an upset) of a partially ordered set

Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...

 (X,≤) is a subset U with the property that x is in U and x≤y imply y is in U.

The dual

Duality (order theory)

In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...

 notion is lower set (alternatively, down set, decreasing set, initial segment; the set is downward closed), which is a subset L with the property that x is in L and y≤x imply y is in L.

Properties

• Every partially ordered set is an upper set of itself.
• The intersection
  Intersection (set theory)
  In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
  
   and the union
  Union (set theory)
  In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
  
   of upper sets is again an upper set.
• The complement
  Complement (set theory)
  In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
  
   of any upper set is a lower set, and vice versa.
• Given a partially ordered set (X,≤), the family of lower sets of X ordered with the inclusion relation is a complete lattice
  Complete lattice
  In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
  
  , the down-set lattice O(X).
• Given an arbitrary subset Y of an ordered set X, the smallest upper set containing Y is denoted using an up arrow as ↑Y.
  • Dually, the smallest lower set containing Y is denoted using a down arrow as ↓Y.
• A lower set is called principal if it is of the form ↓{x} where x is an element of X.
• Every lower set Y of a finite ordered set X is equal to the smallest lower set containing all maximal element
  Maximal element
  In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually...
  
  s of Y: Y = ↓Max(Y) where Max(Y) denotes the set containing the maximal elements of Y.
• A directed
  Directed set
  In mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
  
   lower set is called an order ideal.
• The minimal elements of any upper set form an antichain
  Antichain
  In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two elements in the subset are incomparable. Let S be a partially ordered set...
  
  .
  • Conversely any antichain A determines an upper set {x: for some y in A, x ≥ y}. For partial orders satisfying the descending chain condition this correspondence between antichains and upper sets is 1-1, but for more general partial orders this is not true.

Ordinal numbers

An ordinal number

Ordinal number

In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...

is usually identified with the set of all smaller ordinal numbers. Thus each ordinal number forms a lower set in the class of all ordinal numbers, which are totally ordered by set inclusion.