PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (511)
Corrections (9)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
[parent] difference of lattice elements (Definition)

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\le 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$.



Anyone with an account can edit this entry. Please help improve it!

"difference of lattice elements" is owned by porton. [ full author list (2) ]
(view preamble)

View style:

See Also: complemented lattice, pseudodifference, sectionally complemented lattice


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: bottom, distributive lattice, equation, property, difference, least element, lattice
There are 2 references to this entry.

This is version 7 of difference of lattice elements, born on 2008-03-31, modified 2008-04-08.
Object id is 10464, canonical name is DifferenceOfLatticeElements.
Accessed 312 times total.

Classification:
AMS MSC: 06B99 (Order, lattices, ordered algebraic structures :: Lattices :: Miscellaneous)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy
Existence of no more than one difference by porton on 2008-03-31 13:28:12
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
[ reply | up ]
Interact
post | correct | update request | add derivation | add example | add (any)