Isomorphism between two partially ordered sets.

Question

I want to define an Isomorphism $\phi:\left\langle [n]\times[m],\leq_{Lex}\right\rangle \rightarrow\left\langle [n\cdot m],\leq\right\rangle $

I understand how to write down this isomorphism by hand: lets say $n,m = 2$, then we define:

$(0,0) \rightarrow 0$

$(0,1) \rightarrow 1$

$(1,0) \rightarrow 2$

$(1,1) \rightarrow 3$

However, how could I generalize this idea for arbitrary $n,m$ with an explicit match rule?

Answer 1

It helps a lot if you think about what the lexicographic product means.

$A\times B$ is ordered by replacing each point in $A$ by a copy of $B$. So that means that $n\times m$ means taking $n$ points in a line, and replacing each by $m$ points.

In other words, we look at $n$ multiples of $m$. So a point is identified by identifying which copy, and then its placement within that copy. Or, simply put $$(i,j)\mapsto i\cdot m+j.$$

Answer 2

You may define $$\phi:\left\langle [n]\times[m],\leq_{Lex}\right\rangle \rightarrow\left\langle [n\cdot m],\leq\right\rangle$$ as $$\phi ( i,j)=j+mi$$

For example in case of $ n=3 $ and $m=5$, you have $$\phi (2,3)=3+2(5)=13 $$

$$\phi (1,4)=4+1(5)=9 $$