bounded lattice

A latticeMathworldPlanetmath L is said to be if there is an element 0L such that 0x for all xL. Dually, L is if there exists an element 1L such that x1 for all xL. A bounded latticeMathworldPlanetmath is one that is both from above and below.

For example, any finite lattice L is bounded, as L and L, being join and meet of finitely many elements, exist. L=1 and L=0.

Remarks. Let L be a bounded lattice with 0 and 1 as described above.

  • 0x=0 and 0x=x for all xL.

  • 1x=x and 1x=1 for all xL.

  • As a result, 0 and 1, if they exist, are necessarily unique. For if there is another such a pair 0 and 1, then 0=00=00=0. Similarly 1=1.

  • 0 is called the bottom of L and 1 is called the top of L.

  • L is a lattice interval and can be written as [0,1].

Remark. More generally, a poset P is said to be bounded if it has both a greatest element 1 and a least element 0.

Title bounded lattice
Canonical name BoundedLattice
Date of creation 2013-03-22 15:02:28
Last modified on 2013-03-22 15:02:28
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 06B05
Classification msc 06A06
Defines top
Defines bottom
Defines bounded poset