Upper set

 ( Order Theory ) 

In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X) of a partially ordered set $(X,\; \backslash leq)$ is a subset $S\; \backslash subseteq\; X$ with the following property: if s is in S and if x in X is larger than s (that is, if ), then x is in S. In other words, this means that any x element of X that is $\backslash ,\backslash geq\backslash ,$ 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 $\backslash ,\backslash leq\backslash ,$ to some element of S is necessarily also an element of S.

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

for all $u\; \backslash in\; U$ and all $x\; \backslash in\; X,$ if $u\; \backslash leq\; x$ then $x\; \backslash in\; U.$

The dual notion is a ' (also called a ', ', ', ', or '), which is a subset $L\; \backslash subseteq\; X$ that is "closed under going down", in the sense that

for all $l\; \backslash in\; L$ and all $x\; \backslash in\; X,$ if $x\; \backslash leq\; l$ then $x\; \backslash in\; L.$

The terms ' or ' are sometimes used as synonyms for lower set.

(1998). 9789810233167, World Scientific. . ISBN 9789810233167

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.

(2025). 9780521784511, Cambridge University Press. ISBN 9780521784511

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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 $(X,\; \backslash leq),$ 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 $\backslash uparrow\; Y$ (see upper closure and lower closure).
• Dually, the smallest lower set containing $Y$ is denoted using a down arrow as $\backslash downarrow\; Y.$
• A lower set is called principal if it is of the form $\backslash downarrow\backslash \{x\backslash \}$ 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 of $Y$
• $\backslash downarrow\; Y\; =\; \backslash downarrow\; \backslash operatorname\{Max\}(Y)$ where $\backslash operatorname\{Max\}(Y)$ denotes the set containing the maximal elements of $Y.$
• A Directed set 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 $\backslash \{\; x\; \backslash in\; \backslash R:\; x\; >\; 0\; \backslash \}$ and $\backslash \{\; x\; \backslash in\; \backslash R:\; x\; >\; 1\; \backslash \}$ are both mapped to the empty antichain.

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

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

In this way, $\backslash uparrow\; x\; =\; \backslash uparrow\backslash \{x\backslash \}$ and $\backslash downarrow\; x\; =\; \backslash downarrow\backslash \{x\backslash \},$ 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 Linear span of a set of vectors is the intersection of all Linear subspace 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.)

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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.

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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 $(X,\; \backslash leq)$ that contains for every element $x\; \backslash in\; X,$ some element $y$ such that $x\; \backslash leq\; y.$

• Hoffman, K. H. (2001), The low separation axioms (T₀) and (T₁)

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