Upper set#Upper closure and lower closure

{{short description|Subset of a preorder that contains all larger elements}}

Image:Upset_210div.svg of the divisors of $210$ , ordered by the relation is divisor of, with the upper set $\uparrow 2$ colored green. The white sets form the lower set $\downarrow 105.$ ]]

In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X){{sfn | Dolecki | Mynard | 2016 | pp=27–29}} of a partially ordered set $(X, \leq)$ 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 {{em|upper set}} in $X$ (also called an {{em|upward closed set}}, an {{em|upset}}, or an {{em|isotone}} set){{sfn | Dolecki | Mynard | 2016 | pp=27–29}} 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 {{em|lower set}} (also called a {{em|downward closed set}}, {{em|down set}}, {{em|decreasing set}}, {{em|initial segment}}, or {{em|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 terms {{em|order ideal}} or {{em|ideal}} are sometimes used as synonyms for lower set.{{cite book |last1=Stanley |first1=R.P. |title=Enumerative combinatorics |series=Cambridge studies in advanced mathematics |volume=1 |year=2002 |publisher=Cambridge University Press |isbn=978-0-521-66351-9 | page=100}}{{cite book |last1=Lawson |first1=M.V. |title=Inverse semigroups: the theory of partial symmetries |url=https://archive.org/details/inversesemigroup00laws|url-access=limited |year=1998 |publisher=World Scientific |isbn=978-981-02-3316-7 | page=[https://archive.org/details/inversesemigroup00laws/page/n34 22]}} 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.{{cite book | author1=Brian A. Davey | author2= Hilary Ann Priestley | author2-link= Hilary Priestley | title=Introduction to Lattices and Order|title-link= Introduction to Lattices and Order | edition=2nd | year=2002 | publisher=Cambridge University Press | isbn=0-521-78451-4 | lccn=2001043910 |pages= 20, 44}}

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 $(X, \leq),$ 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.

{{anchor|Upper closure|Upward closure|Lower closure|Downward closure}}

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

$x^{\uparrow X} =\; \uparrow\! x = \{ u \in X : x \leq u\}$

while the lower closure or downward closure of $x$ , denoted by $x^{\downarrow X},$ $x^{\downarrow},$ or $\downarrow\! x,$ is defined by

$x^{\downarrow X} =\; \downarrow\! x = \{l \in X : l \leq x\}.$

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 $A \subseteq X,$ define the upper/upward closure and the lower/downward closure of $A,$ denoted by $A^{\uparrow X}$ and $A^{\downarrow X}$ respectively, as

$A^{\uparrow X} = A^{\uparrow} = \bigcup_{a \in A} \uparrow\!a$

and

$A^{\downarrow X} = A^{\downarrow} = \bigcup_{a \in A} \downarrow\!a.$

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.

References

{{cite journal|author=Blanck, J.|year=2000|title=Domain representations of topological spaces|journal=Theoretical Computer Science|volume=247|issue=1–2|pages=229–255|url=http://www-compsci.swan.ac.uk/~csjens/pdf/top.pdf|doi=10.1016/s0304-3975(99)00045-6|doi-access=free}}
{{Dolecki Mynard Convergence Foundations Of Topology}}
Hoffman, K. H. (2001), [https://web.archive.org/web/20070621125416/http://www.mathematik.tu-darmstadt.de:8080/Math-Net/Lehrveranstaltungen/Lehrmaterial/SS2003/Topology/separation.pdf The low separation axioms (T₀) and (T₁)]

Category:Order theory

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