difference of lattice elements

Let $𝔄$ is a lattice with least element $0$.

Let $a,b\in 𝔄$. A of $a$ and $b$ is an element $c\in 𝔄$ that $b\cap c=0$ and $a\cup b=b\cup c$. When there is only one difference of $a$ and $b$, it is denoted $a\setminus b$.

One immediate property is: $0$ is the unique difference of any element $a$ and itself ($a\setminus a=0$). For if $c$ is such a difference, then $a\cap c=0$ and $a=a\cup c$. So $c\le a$ by the second equation, and hence that $c=a\cap c=0$ 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 lattice 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

• $b\cap c=b\cap d=0$, and

• $a\cup b=b\cup c=b\cup d$.

So $c=c\cap (b\cup c)=c\cap (b\cup d)=(c\cap b)\cup (c\cap d)=0\cup (c\cap d)=c\cap d$. Similarly, $d=d\cap c$. As a result, $c=c\cap d=d\cap c=d$.

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