Let $(A,<_A)$ and $(B,<_B)$ be two partially ordered sets. Show that $(A,<_A)$ is not isomorphic to $(B,<_B)$.

Question

Let $(A,<_A)$ and $(B,<_B)$ be two partially ordered sets, where

$A=\{a,b,c\}$ and $<_A=\{(a,b),(a,c)\}$;

$B=\{x,y,z\}$ and $<_B=\{(x,y),(y,z),(x,z)\}$.

How to show $(A,<_A)$ is not isomorphic to $(B,<_B)$?

Answer 1

Compute maximal elements of $A$ and $B$.

$A$ has $b$ and $c$ as maximal elements. $B$ only has $z$. If they were isomorphic they would have the same number of maximal elements.

Answer 2

If they were, $<_A$ would have the same cardinality as $<_B$.

Answer 3

$B$ is a total (linear) ordering. $A$ is not a linear ordering.

Answer 4

These data are not enough to conclude that the posets are not isomorphic.

You need e.g. the extra condition that $a$, $b$, $c$ are distinct and that $x$, $y$, $z$ are distinct as well. In that case the cardinalities of $<_{A}$ and $<_{B}$ are different which justifies the conclusion that the posets are not isomorphic.