Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I found this definition of alphabetical order for monomials in $k[x_1,\ldots,x_n]$. We say that $x_1^{a_1}\cdots x_n^{a_n}>x_1^{b_1}\cdots x_n^{b_n}$ if for the least $i$ such that $a_i\neq b_i$ we have $a_i>b_i$ and $b_i+\cdots+b_n>0$ or $a_i< b_i$ and $a_{i+1}+\cdots+a_n=0$.

I'm having some problems in understading this definition, for example I want to understand the relation between $x_1$ and $x_1^2$, but it seems to me that with this definition we have both $x_1< x_1^2$ and $x_1^2< x_1$, am I right? what should be the correct definition if this is not correct?

share|cite|improve this question

Alphabetical order sounds a lot like a lexicographical order is meant. That would mean: The alphabetical order on monomials $x_1^{a_1}\ldots x_n^{a_n}$ is the lexicographical order on the sequences $(a_1,a_2,\ldots, a_n)$.

So how exactly to define this? Let $f=x_1^{a_1}\ldots x_n^{a_n}$ and $g=x_1^{b_1}\ldots x_n^{b_n}$. We say that $f<g$ iff for the smallest $i$ where $a_i\neq b_i$, $a_i<b_i$ holds. Just compare this to the lexicon or dictionary: a word is considered 'smaller', or earlier in the lexicon, when on the first place where it differs with another word, the letter comes earlier in the alphabet.

share|cite|improve this answer
is it true that with my definition $x_1<x_1^2$ and $x_1^2<x_1$? – Blu Jan 24 '13 at 1:17

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.