On the alphabetical order of monomial

Question

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?

Answer 1

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.