Sample Page

In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X)^[1] $S$ of a partially ordered set $X$ is a subset such that if s is in S and if x in X is larger than s, that is, if $s<x$ , then x is in S.

A lower set (also called a downward closed set, down set, decreasing set, or semi-ideal) 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.

For a well-ordered set, a lower set is also called an initial segment especially in set theory. An initial segment (but usually not an upper set) is sometimes also defined more generally for a set with a binary relation R like a poset; namely, the same definition but with R in place of <.^[2]

Definition

Let $(X,\leq )$ be a preordered set. An upper set in $X$ (also called an upward closed set, up set, increasing set, or an isotone set)^[1] 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 lower set (also called a downward closed set, down set, decreasing set, initial segment, or a semi-ideal), 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 term order ideal is sometimes used as a synonym for a lower set.^[3]^[4]^[5] However, an ideal is also commonly defined specifically as a lower set which is upward directed.^[6]^[7]

Properties of upper and lower sets

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.
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.
Every upper set $Y$ of a finite partially ordered set $X$ is equal to the smallest upper set containing all minimal elements of $Y.$
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 $\{x\in \mathbb {R} :x>0\}$ and $\{x\in \mathbb {R} :x>1\}$ are both mapped to the empty antichain.

Initial segments of well-ordered sets

Initial segments^[a] are often used in the study of well-ordered sets (posets in which each nonempty subset has a least element) and well-founded sets. For example, an ordinal number is a well-ordered set such that each element $x$ in it is the initial segment

x=\{y\mid y<x\}.

^[8]

Initial segments are also used in the statement of the transfinite recursion theorem.

Properties of initial segments include:

If $X$ is a well-ordered set, then every initial segment of $X$ , other than $X$ itself, is of the form $\{y\mid y<x\}$ for some $x\in X$ .^[9]
A well-ordered set is never isomorphic to a proper initial segment of itself.^[10] Also, given two well-ordered sets $X,Y$ , either $X$ is isomorphic to an initial segment of $Y$ or $Y$ is isomorphic to an initial segment of $X$ .
A morphism between well-ordered sets sends initial segments to initial segments, where a morphism is an injective order-preserving map whose image is an initial segment.^[11]
Initial segments give an ordering $\trianglelefteq$ on the class $\operatorname {Well}$ of well-ordered sets; namely, $A\trianglelefteq B$ if and only if $A\subset B$ is a subset with the ordering restricted from $B$ and $A$ is an initial segment of $B$ .^[12] The union of a chain of well-ordered sets is then well-ordered, where a chain is with respect to $\trianglelefteq$ .^[12] For ordinals, this ordering given by initial segments coincides with set inclusion.^[13]
A set with a binary relation is well-founded if and only if it is covered by well-founded initial segments.^[14]

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 $x^{\uparrow X},$ $x^{\uparrow },$ or $\uparrow \!x,$ is defined by $\uparrow \!x=\{u\in X:x\leq u\}$ while the lower closure or downward closure of $x$ , denoted by $x^{\downarrow X},$ $x^{\downarrow },$ or $\downarrow \!x,$ is defined by $\downarrow \!x=\{l\in X:l\leq x\}.$

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 $A\subset X,$ the upper closure and lower closure of $A$ are defined as $\uparrow A=\bigcup _{a\in A}\uparrow \!a$ and $\downarrow A=\bigcup _{a\in A}\downarrow \!a.$

The upper closure and lower closure of a set are, respectively, the smallest upper set and lower set containing it. Upper and lower sets of the form $\uparrow x$ and $\downarrow x$ are called principal.

The upper and lower closures, when viewed as functions from the power set of $X$ to itself, are examples of Kuratowski closure operators (cf. § Scott topology). 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, a poset can be (and often is) viewed as a category by writing a morphism $x\to y$ if and only if $x\leq y$ . Then the lower closure $\downarrow x$ corresponds to the slice category over $x$ , while the upper closure that under $x$ .^[15]

Let $X$ be a poset.^[16] Then we have^[17]

\eta :X\hookrightarrow {\mathfrak {P}}(X)

where ${\mathfrak {P}}(X)$ is the power set of $X$ and $\eta (x)=\downarrow x$ is the lower closure of $x$ . The map $\eta$ is an embedding in the sense it is injective and monotone

x\leq y\Leftrightarrow \downarrow x\subset \downarrow y.

^[18]

Thus, the above construction can be used to replace a given ordering by set inclusion and also yields advantages such as that a least upper bound always exists (possibly outside the image of $\eta$ ); namely, a union. This trick is used in particular by Halmos in his Naive Set Theory to reduce a proof of Zorn’s lemma to the case of posets of sets.^[19]

As Paul Taylor points out, the above $\eta$ is an analog of an embedding in the Yoneda lemma in category theory.^[20]

Scott topology

A function between posets is said to be Scott-continuous if it is monotone (it preserves $\leq$ ) and preserves directed sups.^[21] Then a poset $X$ carries a topology where a subset $U$ is open if and only if the characteristic function on $U$ is Scott-continuous. This topology is called the Scott topology. Explicitly, an open set in this topology is exactly an upper set such that if $\textstyle \sup _{i}x_{i}\in U$ for a directed set $x_{i}$ , then $x_{i}$ is in $U$ for some $i$ .^[22] 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.

Related notions

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 $x\leq y.$
Filter and ideal – upper and lower sets with the additional property of being directed.

Notes

^ In set theory, the term “initial segment” is commonly used instead of “lower set”.

References

^ ^a ^b Dolecki & Mynard 2016, pp. 27–29.
^ Taylor 1999, Definition 2.6.5.
^ 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.
^ Stanley, R.P. (2002). Enumerative combinatorics. Cambridge studies in advanced mathematics. Vol. 1. Cambridge University Press. p. 100. ISBN 978-0-521-66351-9.
^ Lawson, M.V. (1998). Inverse semigroups: the theory of partial symmetries. World Scientific. p. 22. ISBN 978-981-02-3316-7.
^ Taylor (1999), p. 141: “A directed lower subset of a poset X is called an ideal”
^ Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M. W.; Scott, D. S. (2003). Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications. Vol. 93. Cambridge University Press. p. 3. ISBN 0521803381.
^ Halmos 1960, § 19
^ Tao 2009, Well-ordered sets, Exercise 5.
^ Halmos 1960, § 18
^ Tao 2009, Well-ordered sets, Exercise 7.
^ ^a ^b Halmos 1960, § 17
^ Halmos 1960, § 20.
^ Taylor 1999, Propositon 2.6,6,
^ Taylor 1999, Example 3.1.6. (f)., footnote 1.
^ Editorial note: there is probably a preorder version too.
^ Taylor 1999, Proposition 3.2.7.
^ Taylor 1999, Proposition 3.1.8. (a).
^ Halmos 1960, § 16.
^ Taylor 1999, § 3.1. NB: This exact remark does not appear directly but is clearly implicit in the cited section.
^ Taylor 1999, § 3.4.
^ Taylor 1999, Proposition 3.4.9.

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 2017-08-08. Retrieved 2014-07-21.
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)
Tao, T (28 January 2009). “245B, Notes 7: Well-ordered sets, ordinals, and Zorn’s lemma (optional)”. What’s new.
Halmos, Paul (1960). Naive set theory. Princeton, NJ: D. Van Nostrand Company.. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
Taylor, Paul (13 May 1999). Practical Foundations of Mathematics. Cambridge University Press. ISBN 978-0-521-63107-5. also available at [1]

Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor’s isomorphism theorem Dilworth’s theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal’s tree theorem Laver’s theorem Mirsky’s theorem Szpilrajn extension theorem Zorn’s lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young’s lattice