Upper set

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A Hasse diagram of the power set of the set {1,2,3,4} with the upper set |{1} colored green. The white sets form the lower set |{2,3,4}. Upset.svg
A Hasse diagram of the power set of the set {1,2,3,4} with the upper set ↑{1} colored green. The white sets form the lower set ↓{2,3,4}.

In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X) [1] of a partially ordered set (X, ≤) is a subset SX with the following property: if s is in S and if x in X is larger than s (i.e. if sx), then x is in S. In 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.

Contents

Definition

Let be a preordered set. An upper set in (also called an upward closed set, an upset, or an isotone set) [1] is a subset such that if and if satisfies then That is, satisfies:

for all and all if then

The dual notion is a lower set (also called a downward closed set, down set, decreasing set, initial segment, or semi-ideal), which is a subset such that that if and if satisfies then That is, satisfies:

for all and all if then

The terms order ideal or ideal are sometimes used as synonyms for lower set. [2] [3] [4] 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. [2]

Properties

Upper closure and lower closure

Given an element of a partially ordered set we define the upper closure or upward closure of denoted by or is defined by:

while the lower closure or downward closure of x, denoted by or is defined by:

The sets and are, respectively, the smallest upper and lower sets containing as an element. More generally, given a subset define the upper/upward closure and the lower/downward closures of A, denoted by and respectively, as

     and     

In this way, ↑x = ↑{x} and ↓x = ↓{x}, where upper sets and lower sets of this form are called principal. The upper closures and lower closures of a set are, respectively, the smallest upper set and lower set containing it.

The upper and lower closures, when viewed as function from the power set of X to itself, are examples of closure operators since they satisfy all of the Kuratowski closure axioms. As a result, the upper closure of a set is equal to the intersection of all upper sets containing it, and similarly for lower sets. Indeed, this is a general phenomenon of closure operators. For example, the topological closure of a set is the intersection of all closed sets containing it; the span of a set of vectors is the intersection of all subspaces containing it; the subgroup generated by a subset of a group is the intersection of all subgroups containing it; the ideal generated by a subset of a ring is the intersection of all ideals containing it; and so on.

One can also speak of the strict upper closure of an element defined as {yX : x<y}, and more generally, the strict upper closure of a subset which is defined as the union of the strict upper closures of its elements, and we can make analogous definitions for strict lower closures. However note that these 'closures' are not actually closure operators, since for example the strict upper closure of a singleton set {x} does not contain {x}.

Ordinal numbers

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

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References

  1. 1 2 Dolecki & Mynard 2016, pp. 27–29.
  2. 1 2 Brian A. Davey; Hilary Ann Priestley (2002). Introduction to Lattices and Order (2nd ed.). Cambridge University Press. pp. 20, 44. ISBN   0-521-78451-4. LCCN   2001043910.
  3. Stanley, R.P. (2002). Enumerative combinatorics. Cambridge studies in advanced mathematics. 1. Cambridge University Press. p. 100. ISBN   978-0-521-66351-9.
  4. Lawson, M.V. (1998). Inverse semigroups: the theory of partial symmetries . World Scientific. p.  22. ISBN   978-981-02-3316-7.