Upper set explained

(X, \leq) is a subset S \subseteq X with the following property: if s is in S and if x in X is larger than s (that is, if s < x ), then x is in S. In other words, this means that any x element of X that is \geq 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 \leq to some element of S is necessarily also an element of S.

Definition

Let

(X, \leq) be a preordered set. An in X (also called an , an , or an set) is a subset U \subseteq X that is "closed under going up", in the sense that

for all

u \in U and all x \in X, if u \leq x then x \in U.

The dual notion is a (also called a , , , , or ), which is a subset

L \subseteq X that is "closed under going down", in the sense that

for all

l \in L and all x \in X, if x \leq l then x \in L.

The terms or are sometimes used as synonyms for lower set.^{[1] [2]} 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.^[3]

Properties

• 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

(X, \leq), the family of upper sets of X ordered with the inclusion relation is a complete lattice, the upper set lattice.

• Given an arbitrary subset

Y of a partially ordered set X, the smallest upper set containing Y is denoted using an up arrow as \uparrow Y (see upper closure and lower closure).

• • Dually, the smallest lower set containing

Y is denoted using a down arrow as \downarrow Y.

• A lower set is called principal if it is of the form

\downarrow\{x\} where x is an element of X.

• Every lower set

Y of a finite partially ordered set X is equal to the smallest lower set containing all maximal elements of Y \downarrow Y = \downarrow \operatorname{Max}(Y) where \operatorname{Max}(Y) denotes the set containing the maximal elements of Y.

• A directed lower set is called an order ideal.
• 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

\{ x \in \R: x > 0 \} and \{ x \in \R: x > 1 \} are both mapped to the empty antichain.

Upper closure and lower closure

Given an element

x of a partially ordered set (X, \leq), the upper closure or upward closure of x, denoted by x ^{\uparrow} , x ^{\uparrow} , or \uparrow x, is defined by$$x^\; =\backslash ;\; \backslash uparrow\backslash !\; x\; =\; \backslash$$while the lower closure or downward closure of x , denoted by x ^{\downarrow} , x ^{\downarrow} , or \downarrow x, is defined by$$x^\; =\backslash ;\; \backslash downarrow\backslash !\; x\; =\; \backslash .$$

The sets

\uparrow x and \downarrow x are, respectively, the smallest upper and lower sets containing x as an element. More generally, given a subset A \subseteq X, define the upper/upward closure and the lower/downward closure of A, denoted by A ^{\uparrow} and A ^{\downarrow} respectively, as $$A^\; =\; A^\; =\; \backslash bigcup\_\; \backslash uparrow\backslash !a$$and $$A^\; =\; A^\; =\; \backslash bigcup\_\; \backslash downarrow\backslash !a.$$

In this way,

\uparrow x = \uparrow\{x\} and \downarrow x = \downarrow\{x\}, where upper sets and lower sets of this form are called principal. The upper closure and lower closure of a set are, respectively, the smallest upper set and lower set containing it.

The upper and lower closures, when viewed as functions 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.)

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

• Abstract simplicial complex (also called: Independence system) - a set-family that is downwards-closed with respect to the containment relation.
• Cofinal set – a subset

U of a partially ordered set (X, \leq) that contains for every element x \in X, some element y such that x \leq y.

References

• Blanck, J.. 2000. Domain representations of topological spaces. Theoretical Computer Science. 247. 1–2. 229–255. 10.1016/s0304-3975(99)00045-6. free.
  • Hoffman, K. H. (2001), The low separation axioms (T₀) and (T₁)

Notes and References

1. Book: Stanley . R.P. . Enumerative combinatorics . Cambridge studies in advanced mathematics . 1 . 2002 . Cambridge University Press . 978-0-521-66351-9 . 100.
2. Book: Lawson . M.V. . Inverse semigroups: the theory of partial symmetries . limited . 1998 . World Scientific . 978-981-02-3316-7 . 22.
3. Book: Brian A. Davey . Hilary Ann Priestley . Hilary Priestley . Introduction to Lattices and Order. Introduction to Lattices and Order . 2nd . 2002 . . 0-521-78451-4 . 2001043910 . 20, 44.