Upper set

The powerset algebra of the set {1,2,3,4} with the upset ↑{1} colored green.

In mathematics, an upper set (also called an upward closed set or just an upset) of a partially ordered set (X,≤) is a subset U with the property that, if x is in U and xy, then y is in U.

The dual notion is lower set (alternatively, down set, decreasing set, initial segment, semi-ideal; the set is downward closed), which is a subset L with the property that, if x is in L and yx, then y is in L.

The terms order ideal or ideal are sometimes used as synonyms for lower set.[1][2][3] 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.[1]


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.

See also


  1. 1 2 Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002, p. 20 and 44
  2. Stanley, R.P. (2002). Enumerative combinatorics. Cambridge studies in advanced mathematics. 1. Cambridge University Press. p. 100. ISBN 978-0-521-66351-9.
  3. Lawson, M.V. (1998). Inverse semigroups: the theory of partial symmetries. World Scientific. p. 22. ISBN 978-981-02-3316-7.
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