difference of lattice elements
Let $a,b\in \U0001d504$. A of $a$ and $b$ is an element $c\in \U0001d504$ 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

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$b\cap c=b\cap d=0$, and

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$a\cup b=b\cup c=b\cup d$.
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  20130322 17:57:44 
Last modified on  20130322 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 