Upper set

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

${\displaystyle a\in S\land b\geq a\implies b\in S}${\displaystyle a\in S\land b\geq a\implies b\in 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 ${\displaystyle ,円\leq ,円}${\displaystyle ,円\leq ,円} to some element of S is necessarily also an element of S.

Let ${\displaystyle (X,\leq )}${\displaystyle (X,\leq )} be a preordered set. An upper set in ${\displaystyle X}${\displaystyle X} (also called an upward closed set, up set, increasing set, or an isotone set)^[1] is a subset ${\displaystyle U\subseteq X}${\displaystyle U\subseteq X} that is "closed under going up", in the sense that

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

The dual notion is a lower set (also called a downward closed set, down set, decreasing set, initial segment, or a semi-ideal), which is a subset ${\displaystyle L\subseteq X}${\displaystyle L\subseteq X} that is "closed under going down", in the sense that

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

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]

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

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

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

In this way, ${\displaystyle \uparrow x=\uparrow \{x\}}${\displaystyle \uparrow x=\uparrow \{x\}} and ${\displaystyle \downarrow x=\downarrow \{x\},}${\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 ${\displaystyle X}${\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.)

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.

• Abstract simplicial complex (also called: Independence system) - a set-family that is downwards-closed with respect to the containment relation.
• Cofinal set – a subset ${\displaystyle U}${\displaystyle U} of a partially ordered set ${\displaystyle (X,\leq )}${\displaystyle (X,\leq )} that contains for every element ${\displaystyle x\in X,}${\displaystyle x\in X,} some element ${\displaystyle y}${\displaystyle y} such that ${\displaystyle x\leq y.}${\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 . Archived from the original (PDF) on 2017年08月08日. Retrieved 2014年07月21日.
• 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)

• Order theory

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