criteria for a poset to be a complete lattice

PropositionPlanetmathPlanetmathPlanetmath. Let L be a poset. Then the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

  1. 1.
  2. 2.

    for every subset A of L, A exists.

  3. 3.

    for every finite subset F of L and every directed setMathworldPlanetmath D of L, F and D exist.


ImplicationsMathworldPlanetmath 1.2.3. are clear. We will show 3.2.1.

(3.2.) If A=, then A=0 by definition. So assume A be a non-empty subset of L. Let A be the set of all finite subsets of A and B={FFA}. By assumptionPlanetmathPlanetmath, B is well-defined and AB. Next, let B be the set of all directed subsets of B, and C={DDB}. By assumption again, C is well-defined and BC. Now, every chain in C has a maximal elementMathworldPlanetmath in C (since a chain is a directed set), C itself has a maximal element d by Zorn’s Lemma. We will show that d is the least upper boundMathworldPlanetmath of elments of A. It is clear that each aA is bounded above by d (ABC). If t is an upper bound of elements of A, then it is an upper bound of elements of B, and hence an upper bound of elements of C, which means dt.

(2.1.) By assumption exists (=0), so that L=0. Now suppose A is a proper subsetMathworldPlanetmathPlanetmath of L. We want to show that A exists. If A=, then A=L=1 by definition of an arbitrary meet over the empty setMathworldPlanetmath. So assume A. Let A be the set of lower bounds of A: A={xLxa for all aA} and let b=A, the least upper bound of A. b exists by assumption. Since A is a set of upper bounds of A, ba for all aA. This means that b is a lower bound of elements of A, or bA. If x is any lower bound of elements of A, then xb, since x is bounded above by b (b=A). This shows that A exists and is equal to b. ∎


  • Dually, a poset is a complete lattice iff every subset has an infimumMathworldPlanetmathPlanetmath iff infimum exists for every finite subset and every directed subset.

  • The above proposition shows, for example, that every closure system is a complete lattice.

Title criteria for a poset to be a complete lattice
Canonical name CriteriaForAPosetToBeACompleteLattice
Date of creation 2013-03-22 16:37:53
Last modified on 2013-03-22 16:37:53
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Theorem
Classification msc 06B23
Classification msc 03G10
Related topic MeetContinuous
Related topic IntersectionStructure