Give counterexample to the following claim about invertible homomorphic posets

Question

In general, it is known that all invertible (bijective) group homomorphisms are group isomorphisms. However the same reasoning need not hold true for invertible poset homomorphisms. That is, not all invertible poset homomorphisms are poset isomorphisms. I'm having trouble coming up with an example that validates this claim - since I'm relatively new to Abstract Algebra. Any help would be appreciated.

Answer 1

"In general, it is known that all invertible (bijective) group homomorphisms are group isomorphisms"

This is true for sets and groups, but not for abritrary objects on a category. For example, not all bijective continuous functions have a continuous inverse.

As for the posets counterexample: let $a \to b$ be a poset of two elements with $a \leq b$, and $\{c,d\}$ be a poset where no elements are comparable. Let $f(a) = c, f(b) = d$ and $g(c) = a, g(d) = b$ be inverse functions. In particular, both are bijective. Since $g$ is a homomorphism (the poset has no comparable objects so the condition is satisfied trivially), by your claim its inverse should be a morphism, but $f$ is not: $a \leq b$ would imply $f(a) = c \leq d = f(b)$, but once again, $c$ and $d$ are not comparable.