difference of lattice elements

Let 𝔄 is a latticeMathworldPlanetmath with least element 0.

Let a,b𝔄. A of a and b is an element c𝔄 that bc=0 and ab=bc. When there is only one differencePlanetmathPlanetmath of a and b, it is denoted ab.

One immediate property is: 0 is the unique difference of any element a and itself (aa=0). For if c is such a difference, then ac=0 and a=ac. So ca by the second equation, and hence that c=ac=0 by the first equation.

For arbitrary lattices of two given elements do not necessarily exist. For some lattices there may be more than one difference of two given elements.

For a distributive latticeMathworldPlanetmath with bottom element 0, the difference of two elements, if it exists, must be unique. To see this, let c and d be two differences of a and b. Then

  • bc=bd=0, and

  • ab=bc=bd.

So c=c(bc)=c(bd)=(cb)(cd)=0(cd)=cd. Similarly, d=dc. As a result, c=cd=dc=d.

Title difference of lattice elements
Canonical name DifferenceOfLatticeElements
Date of creation 2013-03-22 17:57:44
Last modified on 2013-03-22 17:57:44
Owner porton (9363)
Last modified by porton (9363)
Numerical id 10
Author porton (9363)
Entry type Definition
Classification msc 06B99
Related topic ComplementedLattice
Related topic Pseudodifference
Related topic SectionallyComplementedLattice