In lattice homomorphism $f: L \to M$, does $f(a)\leq f(b)$ necessarily imply $a\leq b$?
if $a \leq b$ does not hold. Then $b\leq a$ (because $f(a)$ and $f(b)$ are comparable so $a$ and $b$ have to be comparable somehow ?)
$a\wedge b \leq a \vee b$ $\to$ $f(a\wedge b) \leq f(a \vee b)$ $\to f(b) \leq f(a)$ contradiction.
Sorry I am new to discrete maths. Please help me