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Homedifference of lattice elements
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difference of lattice elements
Let $\mathfrak{A}$ is a lattice with least element $0$.
Let $a,b\in\mathfrak{A}$. A difference of $a$ and $b$ is an element $c\in\mathfrak{A}$ 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 $c\leq a$ by the second equation, and hence that $c=a\cap c=0$ by the first equation.
For arbitrary lattices differences 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, $c=c\cap d=d\cap c=d$.
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Existence of no more than one difference
How to prove the following theorem (or something similar):
If our lattice is distributive, then exist no more than one difference of given two elements.
There exists an similar theorem saying that for distributive lattice there exists no more than one complement of given element, but I need the more general case of difference instead of complement.
See also:
http://planetmath.org/encyclopedia/ComplementedLattice.html

Victor Porton  http://www.mathematics21.org
* Algebraic General Topology and Math Synthesis
* Category Theory  new concepts
Re: Existence of no more than one difference
How would you define the difference of two elements in a distributive lattice?
Re: Existence of no more than one difference
Never mind, I just saw it.
Re: Existence of no more than one difference
I have provided a proof of this in your entry. Please take a look.