Math 2.0

Order, lattices, ordered algebraic structures Definition

Conditionally complete lattice

Builds on LatticeUpper boundSupremumInfimumReal numbersComplete lattice

Definition

Let (L,≤)(L, \le) be a lattice. It is a conditionally complete lattice if:

  1. every nonempty subset of LL that has an upper bound in LL has a supremum in LL, and
  2. every nonempty subset of LL that has a lower bound in LL has an infimum in LL.

Remarks

Clause 2 follows from clause 1. Indeed, the infimum of a nonempty set SS bounded below is the supremum of the set of lower bounds of SS. That set is nonempty and bounded above by any element of SS. Example: the real numbers with the usual order, by the completeness axiom. Non-example: ℚ\Q, since {q∈ℚ:q2<2}\set{q \in \Q : q^2 < 2} is nonempty and bounded above by 22 but has no supremum in ℚ\Q. Every complete lattice is conditionally complete, and a conditionally complete lattice with a least and a greatest element is complete. A totally ordered set with property 1 is said to have the least-upper-bound property.

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Complete lattice, concept, no evidence levelComplete latticeconceptInfimum, concept, no evidence levelInfimumconceptLattice, concept, no evidence levelLatticeconceptReal numbers, object, no evidence levelReal numbersobjectSupremum, concept, no evidence levelSupremumconceptUpper bound, concept, no evidence levelUpper boundconceptConditionally complete lattice, definition, no evidence levelConditionally complet…definition
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  • no evidence level
  1. Complete lattice concept
  2. Infimum concept
  3. Lattice concept
  4. Real numbers object
  5. Supremum concept
  6. Upper bound concept
  7. Conditionally complete lattice definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Completeness_(order_theory)
About this page
TagA5D94C38HZ
URIm2://A5D94C38HZ
Kinddefinition
WritingW0 stub
MSC06B23
DefinesConditionally complete lattice
UsesLattice, Upper bound, Supremum, Infimum, Real numbers, Complete lattice
Changed2 November 2026

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