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Wikipedia / Upper and lower sets

In mathematics, an upper set of a partially ordered set is a subset such that if s is in S and if x in X is larger than s, then x is in S. A lower set is defined similarly as being a subset S of X with the property that any element x of X that precedes an element of S is necessarily also an element of S.

Upper sets and lower sets are also known by many other names. An upper set may also be called an upward closed set, an up-set, an isotone set, or an order filter, while a lower set may also be called a downward closed set, down-set, decreasing set, semi-ideal, or order ideal. However, the terms "order ideal" and "order filter" are also used for a more restrictive notion.

Definition

be a preordered set is dropped).

An (also called an , , , or an ).

The dual notion is a (also called a , , , or a

The term is sometimes used as a synonym for a lower set. However, an ideal is also commonly defined specifically as a lower set which is upward directed. Dually, a filter is an upper set that is directed downward (that is, every finite subset has a lower bound).

For a well-ordered set, a lower set is usually called an initial segment.

Properties

The following properties are stated in terms of upper sets; the corresponding dual properties for lower sets also hold.

  • Every preordered 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 an upper set is a lower set, and vice versa.

  • ordered with the inclusion relation is a complete lattice, the upper set lattice.

  • is equal to the smallest upper set containing all minimal elements

  • For partial orders satisfying the descending chain condition, antichains and upper sets are in one-to-one correspondence via the following bijections: map each antichain to its upper closure (see below); conversely, map each upper set to the set of its minimal elements. This correspondence does not hold for more general partial orders; for example the sets of real numbers are both mapped to the empty antichain.

Examples

Upper sets and lower sets appear in various fields of mathematics.

  • In the totally ordered set of the real numbers.

  • In real analysis, a real number is often defined as a Dedekind cut with no maximal element.

  • is an upper set in the power set ordered by inclusion, since any set containing a neighborhood of the point is a neighborhood of that point.

  • Any filter on a set ordered by inclusion. The previous example of the neighbourhood filter of a point in a topological space is an instance of this.

  • Abstract simplicial complex - a set-family that is downwards-closed with respect to the containment relation.

Upper closure and lower closure

the upper closure or upward closure is defined by

while the lower closure or downward closure by

are called principal. The upper closure of an element is the same thing as the principal filter generated by that element, since it is also directed downward.

; to itself, are examples of Kuratowski closure operators. 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.

In category theory corresponds to the slice category.

be a poset. Then we have

is an embedding in the sense it is injective and monotone:

); namely, a union. For example, this trick can be used to reduce a proof of Zorn's lemma to the case of posets of sets.

is an analog of an embedding in the Yoneda lemma in category theory.

. But, more specifically, it lies in the set of all directed and called the ideal completion. a free functor in the sense: it is left adjoint to the forgetful functor from the category of dcposs to the category of posets.

Scott topology

A function between posets is said to be Scott-continuous) and preserves directed sups. is Scott-continuous. This topology is called the Scott topology. The intuition here is that a sup corresponds to the best approximation and so if the best approximation is available in the set, some finite approximation is already in that set.

The Scott topology appears prominently in domain theory, a branch of order theory with a strong connection to computer science. Like the Zariski topology used in algebraic geometry, the Scott topology is an important example of a non-Hausdorff topological space.

Birkhoff's theorem

, is a lattice has a least upper bound and a greatest lower bound), and indeed a distributive lattice (meaning that the two operations of least upper bound and greatest lower bound distribute over one another). Birkhoff's representation theorem asserts that every finite distributive lattice arises (up to isomorphism) in this way as the lattice of lower sets of a unique finite poset.

Related notions

See also

Notes

References

ru:Частично упорядоченное множество#Верхнее и нижнее множество