lattice interval

Definition. Let L be a latticeMathworldPlanetmath. A subset I of L is called a lattice interval, or simply an if there exist elements a,bL such that


The elements a,b are called the endpoints of I. Clearly a,bI. Also, the endpoints of a lattice interval are unique: if [a,b]=[c,d], then a=c and b=d.


  • It is easy to see that the name is derived from that of an interval on a number line. From this analogyMathworldPlanetmath, one can easily define lattice intervals without one or both endpoints. Whereas an interval on a number line is linearly orderedPlanetmathPlanetmath, a lattice interval in general is not. Nevertheless, a lattice interval I of a lattice L is a sublattice of L.

  • A bounded latticeMathworldPlanetmath is itself a lattice interval: [0,1].

  • A prime interval is a lattice interval that contains its endpoints and nothing else. In other words, if [a,b] is prime, then any c[a,b] implies that either c=a or c=b. Simply put, b covers a. If a lattice L contains 0, then for any aL, [0,a] is a prime interval iff a is an atom.

  • Since no operationsMathworldPlanetmath of meet and join are used, all of the above discussion can be generalized to define an interval in a poset.

  • Given a lattice L, let be the collectionMathworldPlanetmath of all lattice intervals without endpoints, we can form a topolgy on L with as the subbasis. This does not insure that and are continuousMathworldPlanetmathPlanetmath, so that L with this topological structure may not be a topological lattice.

  • Locally Finite Lattice. A lattice that is derived based on the concept of lattice interval is that of a locally finite lattice. A lattice L is locally finitePlanetmathPlanetmathPlanetmath iff every one of its interval is finite. Unless the lattice is finite, a locally finite lattice, if infiniteMathworldPlanetmath, is either topless or bottomless.

Title lattice interval
Canonical name LatticeInterval
Date of creation 2013-03-22 15:44:56
Last modified on 2013-03-22 15:44:56
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 06B99
Classification msc 06A06
Defines prime interval
Defines poset interval
Defines locally finite lattice