Difference between product order and lexicographic order

Question

I know the definitions of these two terms, but I saw this question in one of my former exams from "Discrete math". The question stated :

"What is the difference between product order and lexicographic order, and give an example that shows the difference?"

Could someone clear this out for me, I would be very grateful.

Answer 1

Suppose that $\le$ is a linear order on a set $X$. The lexicographic order on $X\times X$ is a linear order; you should try to prove this if you’ve not already done so or seen a proof of it. The product order on $X\times X$ is not a linear order, provided that $X$ has at least two elements. To see this, let $x$ and $y$ be distinct elements of $X$, and let $\le_p$ be the product order on $X\times X$. Then $\langle x,y\rangle\not\le_p\langle y,x\rangle$, and $\langle y,x\rangle\not\le_p\langle x,y\rangle$, so the pairs $\langle x,y\rangle$ and $\langle y,x\rangle$ are not comparable with respect to $\le_p$.

Answer 2

Take the usual order on integers. In the lexicographical order: $\langle 1,4\rangle < \langle 2,4 \rangle$. In the product order these two elements are not comparable.