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 is a subset 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 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 to some element of S is necessarily also an element of S.

A Hasse diagram of the divisors of , ordered by the relation is divisor of, with the upper set colored green. The white sets form the lower set

Definition

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Let   be a preordered set. An upper set in   (also called an upward closed set, an upset, or an isotone set)[1] is a subset   that is "closed under going up", in the sense that

for all   and all   if   then  

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   that is "closed under going down", in the sense that

for all   and all   if   then  

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]

Properties

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  • Every partially ordered 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   the family of upper sets of   ordered with the inclusion relation is a complete lattice, the upper set lattice.
  • Given an arbitrary subset   of a partially ordered set   the smallest upper set containing   is denoted using an up arrow as   (see upper closure and lower closure).
    • Dually, the smallest lower set containing   is denoted using a down arrow as  
  • A lower set is called principal if it is of the form   where   is an element of  
  • Every lower set   of a finite partially ordered set   is equal to the smallest lower set containing all maximal elements of  
    •   where   denotes the set containing the maximal elements of  
  • 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   and   are both mapped to the empty antichain.

Upper closure and lower closure

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Given an element   of a partially ordered set   the upper closure or upward closure of   denoted by     or   is defined by   while the lower closure or downward closure of  , denoted by     or   is defined by  

The sets   and   are, respectively, the smallest upper and lower sets containing   as an element. More generally, given a subset   define the upper/upward closure and the lower/downward closure of   denoted by   and   respectively, as   and  

In this way,   and   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   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

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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

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  • Abstract simplicial complex (also called: Independence system) - a set-family that is downwards-closed with respect to the containment relation.
  • Cofinal set – a subset   of a partially ordered set   that contains for every element   some element   such that  

References

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  1. ^ a b Dolecki & Mynard 2016, pp. 27–29.
  2. ^ a b 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.
  3. ^ Stanley, R.P. (2002). Enumerative combinatorics. Cambridge studies in advanced mathematics. Vol. 1. Cambridge University Press. p. 100. ISBN 978-0-521-66351-9.
  4. ^ Lawson, M.V. (1998). Inverse semigroups: the theory of partial symmetries. World Scientific. p. 22. ISBN 978-981-02-3316-7.