upper set

Let P be a poset and A a subset of P. The upper set of A is defined to be the set

{bPab for some aA},

and is denoted by A. In other words, A is the set of all upper boundsMathworldPlanetmath of elements of A.

can be viewed as a unary operator on the power setMathworldPlanetmath 2P sending A2P to A2P. has the following properties

  1. 1.


  2. 2.


  3. 3.

    A=A, and

  4. 4.

    if AB, AB.

So is a closure operatorPlanetmathPlanetmathPlanetmath.

An upper set in P is a subset A such that its upper set is itself: A=A. In other words, A is closed with respect to in the sense that if aA and ab, then bA. An upper set is also said to be upper closed. For this reason, for any subset A of P, the A is also called the upper closure of A.

Dually, the lower set (or lower closure) of A is the set of all lower bounds of elements of A. The lower set of A is denoted by A. If the lower set of A is A itself, then A is a called a lower set, or a lower closed set.


  • A is not the same as the set of upper bounds of A, commonly denoted by Au, which is defined as the set {bPab for all aA}. Similarly, AA in general, where A is the set of lower bounds of A.

  • When A={x}, we write x for A and x for A. x={x}u and x={x}d.

  • If P is a latticeMathworldPlanetmath and xP, then x is the principal filterPlanetmathPlanetmathPlanetmath generated by x, and x is the principal idealPlanetmathPlanetmath generated by x.

  • If A is a lower set of P, then its set complementPlanetmathPlanetmath A is an upper set: if aA and ab, then bA by a contrapositive argumentMathworldPlanetmath.

  • Let P be a poset. The set of all lower sets of P is denoted by 𝒪(P). It is easy to see that 𝒪(P) is a poset (ordered by inclusion), and 𝒪(P)=𝒪(P), where is the dualization operationMathworldPlanetmath (meaning that P is the dual poset of P).

Title upper set
Canonical name UpperSet
Date of creation 2013-03-22 15:49:50
Last modified on 2013-03-22 15:49:50
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 20
Author CWoo (3771)
Entry type Definition
Classification msc 06A06
Synonym up set
Synonym down set
Synonym upper closure
Synonym lower closure
Related topic LatticeIdeal
Related topic LatticeFilter
Related topic Filter
Defines lower set
Defines upper closed
Defines lower closed