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: Very high
lexicographic order (Definition)

Let $ A$ be a set equipped with a total order $ <$, and let $ A^n=A\times \cdots \times A$ be the $ n$-fold Cartesian product of $ A$. Then the lexicographic order $ <$ on $ A^n$ is defined as follows:

If $ a=(a_1, \ldots, a_n)\in A^n$ and $ b=(b_1, \ldots, b_n)\in A^n$, then $ a<b$ if $ a_1<b_1$ or

$\displaystyle a_1$ $\displaystyle =$ $\displaystyle b_1,$  
  $\displaystyle \vdots$    
$\displaystyle a_k$ $\displaystyle =$ $\displaystyle b_k,$  
$\displaystyle a_{k+1}$ $\displaystyle <$ $\displaystyle b_{k+1}$  

for some $ k=1,\ldots, n-1$.

Examples

  • The lexicographic order yields a total order on the field of complex numbers.
  • The lexicographic order of words of finite length consisting of letters $ '\ \ '$ (space) $ <a<b<\cdots<y<z$ is the dictionary order. To compare words of different length, one simply pads the shorter with $ '\ \ '$s from the right. For example, $ \operatorname{prove} < \operatorname{proved} < \operatorname{proven}$.

Properties



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

"lexicographic order" is owned by matte. [ full author list (4) ]
(view preamble)

View style:

Also defines:  dictionary order
Log in to rate this entry.
(view current ratings)

Cross-references: well-ordering, product, lexicographic ordering, well-ordered, right, length, finite length, complex numbers, field, Cartesian product, total order
There are 9 references to this entry.

This is version 10 of lexicographic order, born on 2005-05-04, modified 2006-10-16.
Object id is 7005, canonical name is LexicographicOrder.
Accessed 4493 times total.

Classification:
AMS MSC: 06A99 (Order, lattices, ordered algebraic structures :: Ordered sets :: Miscellaneous)
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)