Let $X=\{1,2,3,4\}$. Are the monoids $(X,\max), (X,\min)$ isomorphic?

Question

Let $X=\{1,2,3,4\}$. How do I make sure if the monoids $(X,\max), (X,\min)$ are isomorphic? I can see that it's possible to define a bijective function between these two, but can this function also be homomorphic?

Answer 1

More generally let $X=\{0, 1,\ldots, n\}$ (note that I've added $0$) and define

$$f:X\to X$$ $$f(k)=n-k$$

Simply speaking $f$ reverses the order.

Then for any $a, b\in X$ we have

$$f\big(\max(a, b)\big)=n - \max(a,b)=\min(n-a, n-b)=\min\big(f(a), f(b)\big)$$ $$f\big(\min(a, b)\big)=n - \min(a,b)=\max(n-a, n-b)=\max\big(f(a), f(b)\big)$$

Those equalities follow from the fact that $a\leq b$ if and only if $-a\geq -b$.

So $f$ preserves both monoid structures and is invertible with $f^{-1}=f$. Indeed

$$f(f(k))=f(n-k)=n-(n-k)=k$$

Thus it is an isomorphism.