I am reading about about lexicographic ordering, and I want to make sure I am understanding it properly. Lexicographic ordering is defined to be the cartesian product of two, or more, posets. So, $A_1 = (B_1, \preceq)$, and $A_2 = (B_2, \preceq)$; and $A_1 \times A_2$ is a lexicographic order? And $((a,b),(c,d))$ is in $A_1 \times A_2$, if $(a,b)$ is in $A_1$, and $(c,d)$ is in $A_2$. But for the cartesian product to be defined, it seems like the relation in each poset has to be the same. Is that right?
A part of what the book says, that I don't understand, is, "Lexicographic orderingis defined by specifying that one pair is less than a second pair if the first entry of the first pair is less than (inA1) the first entry of the second pair, or if the first entries are equal, but the second entry of this pair is less than (inA2) the second entry of the second pair. In other words,(a1,a2)is less than(b1,b2), that is, (a1,a2)≺(b1,b2), either if $a1≺b1$ or if both $a_1=b_1$ and $a_2≺b_2$."