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Definition

Let ( X , ≤ ) {\displaystyle (X,\leq )} be a preordered set. An upper set in X {\displaystyle X} (also called an upward closed set, an upset, or an isotone set)² is a subset U ⊆ X {\displaystyle U\subseteq X} that is "closed under going up", in the sense that

for all u ∈ U {\displaystyle u\in U} and all x ∈ X , {\displaystyle x\in X,} if u ≤ x {\displaystyle u\leq x} then x ∈ U . {\displaystyle x\in U.}

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 L ⊆ X {\displaystyle L\subseteq X} that is "closed under going down", in the sense that

for all l ∈ L {\displaystyle l\in L} and all x ∈ X , {\displaystyle x\in X,} if x ≤ l {\displaystyle x\leq l} then x ∈ L . {\displaystyle x\in L.}

The terms order ideal or ideal are sometimes used as synonyms for lower set.³⁴⁵ 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.⁶

Properties

• 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 any upper set is a lower set, and vice versa.
• Given a partially ordered set ( X , ≤ ) , {\displaystyle (X,\leq ),} the family of upper sets of X {\displaystyle X} ordered with the inclusion relation is a complete lattice, the upper set lattice.
• Given an arbitrary subset Y {\displaystyle Y} of a partially ordered set X , {\displaystyle X,} the smallest upper set containing Y {\displaystyle Y} is denoted using an up arrow as ↑ Y {\displaystyle \uparrow Y} (see upper closure and lower closure).
  • Dually, the smallest lower set containing Y {\displaystyle Y} is denoted using a down arrow as ↓ Y . {\displaystyle \downarrow Y.}
• A lower set is called principal if it is of the form ↓ { x } {\displaystyle \downarrow \{x\}} where x {\displaystyle x} is an element of X . {\displaystyle X.}
• Every lower set Y {\displaystyle Y} of a finite partially ordered set X {\displaystyle X} is equal to the smallest lower set containing all maximal elements of Y {\displaystyle Y}
  • ↓ Y =↓ Max ⁡ ( Y ) {\displaystyle \downarrow Y=\downarrow \operatorname {Max} (Y)} where Max ⁡ ( Y ) {\displaystyle \operatorname {Max} (Y)} denotes the set containing the maximal elements of Y . {\displaystyle 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 ∈ R : x > 0 } {\displaystyle \{x\in \mathbb {R} :x>0\}} and { x ∈ R : x > 1 } {\displaystyle \{x\in \mathbb {R} :x>1\}} are both mapped to the empty antichain.

Upper closure and lower closure

Given an element x {\displaystyle x} of a partially ordered set ( X , ≤ ) , {\displaystyle (X,\leq ),} the upper closure or upward closure of x , {\displaystyle x,} denoted by x ↑ X , {\displaystyle x^{\uparrow X},} x ↑ , {\displaystyle x^{\uparrow },} or ↑ x , {\displaystyle \uparrow \!x,} is defined by x ↑ X = ↑ x = { u ∈ X : x ≤ u } {\displaystyle x^{\uparrow X}=\;\uparrow \!x=\{u\in X:x\leq u\}} while the lower closure or downward closure of x {\displaystyle x} , denoted by x ↓ X , {\displaystyle x^{\downarrow X},} x ↓ , {\displaystyle x^{\downarrow },} or ↓ x , {\displaystyle \downarrow \!x,} is defined by x ↓ X = ↓ x = { l ∈ X : l ≤ x } . {\displaystyle x^{\downarrow X}=\;\downarrow \!x=\{l\in X:l\leq x\}.}

The sets ↑ x {\displaystyle \uparrow \!x} and ↓ x {\displaystyle \downarrow \!x} are, respectively, the smallest upper and lower sets containing x {\displaystyle x} as an element. More generally, given a subset A ⊆ X , {\displaystyle A\subseteq X,} define the upper/upward closure and the lower/downward closure of A , {\displaystyle A,} denoted by A ↑ X {\displaystyle A^{\uparrow X}} and A ↓ X {\displaystyle A^{\downarrow X}} respectively, as A ↑ X = A ↑ = ⋃ a ∈ A ↑ a {\displaystyle A^{\uparrow X}=A^{\uparrow }=\bigcup _{a\in A}\uparrow \!a} and A ↓ X = A ↓ = ⋃ a ∈ A ↓ a . {\displaystyle A^{\downarrow X}=A^{\downarrow }=\bigcup _{a\in A}\downarrow \!a.}

In this way, ↑ x =↑ { x } {\displaystyle \uparrow x=\uparrow \{x\}} and ↓ x =↓ { x } , {\displaystyle \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 {\displaystyle 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 {\displaystyle U} of a partially ordered set ( X , ≤ ) {\displaystyle (X,\leq )} that contains for every element x ∈ X , {\displaystyle x\in X,} some element y {\displaystyle y} such that x ≤ y . {\displaystyle x\leq y.}

• Blanck, J. (2000). "Domain representations of topological spaces" (PDF). Theoretical Computer Science. 247 (1–2): 229–255. doi:10.1016/s0304-3975(99)00045-6.
• Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
• Hoffman, K. H. (2001), The low separation axioms (T0) and (T1)

References

1. Dolecki & Mynard 2016, pp. 27–29. - Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917. https://search.worldcat.org/oclc/945169917 ↩

2. Dolecki & Mynard 2016, pp. 27–29. - Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917. https://search.worldcat.org/oclc/945169917 ↩

3. 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. 0-521-78451-4 ↩

4. Stanley, R.P. (2002). Enumerative combinatorics. Cambridge studies in advanced mathematics. Vol. 1. Cambridge University Press. p. 100. ISBN 978-0-521-66351-9. 978-0-521-66351-9 ↩

5. Lawson, M.V. (1998). Inverse semigroups: the theory of partial symmetries. World Scientific. p. 22. ISBN 978-981-02-3316-7. 978-981-02-3316-7 ↩

6. 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. 0-521-78451-4 ↩