Let $X=\{x,y\}$ and $ x>y $ lexicographically. Let consider $X^*$ as a free monoid generated by $X$. We define a monomial ordering on $X^*$ which is compatible with the multiplication of words, due to its definition. Let $deg-lex$ be the ordering defined on $X^*$.
Then I want to order the following (all monomials of degree 3) by $deg-lex$ ordering. I just want to ask you about the correctness of the way I have already ordered them?
$y^{3} < y^{2}x< yxy <yx^{2} <xy^{2} <xyx<x^{2}y <x^{3}$
Your answer looks correct.
A bit of terminology. Since you refer to a free monoid, I would call your order the shortlex order, also known as radix, length-lexicographic, military, or genealogical ordering. The term deg-lex is rather used for monomials of a polynomial ring.