Upper set

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The powerset algebra of the set {1,2,3,4} with the upset ↑{1} colored green.

In mathematics, an upper set (also called an upward closed set or just an upset) of a partially ordered set (X,≤) is a subset U with the property that, if x is in U and xy, then y is in U.

The dual 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, if x is in L and yx, then y is in L.

Properties[edit]

  • Every partially ordered set is an upper set of itself.
  • The intersection and the union 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 (X,≤), the family of lower sets of X ordered with the inclusion relation is a complete lattice, 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 elements of Y: Y = ↓Max(Y) where Max(Y) denotes the set containing the maximal elements of Y.
  • A directed lower set is called an order ideal.
  • The minimal elements of any upper set form an antichain.
    • Conversely any antichain A determines an upper set {x: for some y in A, xy}. 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[edit]

An ordinal number 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.

See also[edit]

  • Cofinal set – a subset U of a partially ordered set (P,≤) that contains for every element x of P an element y such that xy

References[edit]