On the alphabetical order of monomial

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?

-

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.
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