AoPSWiki
Looking for a challenging geometry text? Preparing for MATHCOUNTS or the AMC exams? Check out Art of Problem Solving's Introduction to Geometry by Richard Rusczyk.

Lexicographic ordering

From AoPSWiki

Lexicographic (or dictionary ordering) is a strict total ordering < on the \alpha-tuple A = \prod_{\beta < \alpha} A_{\beta} of totally ordered sets for some ordinal \alpha using the strict total orderings <_{\beta} of A_{\beta}, \beta < \alpha.

Formal Definition

For (a,b) \in A^2, letting a_{\beta} and b_{\beta} denote the \beta^{\text{th}} components of a and b respectively, we say a < b iff (\exists \gamma \leq \alpha)((\forall \beta < \gamma)(a_{\beta} = b_{\beta}) \wedge (a_{\gamma} <_{\gamma} b_{\gamma})).

Do you have what it takes to be the next brilliant trader, researcher, or developer at Jane Street Capital? Find out in the Careers in Mathematics Forum.
© Copyright 2008 AoPS Incorporated. All Rights Reserved. • FoundationPrivacyContact Us