I read on Applied Abstract Algebra by Rudolf Lidl and Gunter Pilz that if $L$ and $M$ be two lattices and $f:L\to M$ be a mapping, then $f$ is order homomorphism if $x\le y\implies f(x)\le f(y)$.
On the other hand, in Introduction to Lattices and Order by B.A.Davey and H.A.Priestley, it is said that $f$ is order isomomorphism if $x\le y\iff f(x)\le f(y)$.
Now these two are confusing me whether they are same or if there is any difference, is it just only 'if' and 'iff'?
Also if I am asked to show that $f$ is a bijective homomorphism (is it equal to isomorphism?), how do I show that? Because my knowledge says that to show bijection , I need to show both injection and surjection. And for homomorphism, I need to show join and meet-homomorphism. Here my second question arises how to show injective and surjective in this case? Is it same as injective and surjective mapping?
Note that if $f$ is an order isomorphism, then $f$ must be one-to-one.
Indeed, if $f(x)=f(y)$ then $f(x) \leq f(y)$ implies $x \leq y$, while $f(y) \leq f(x)$ implies $y \leq x$. Therefore $x=y$.
The same is not true for order homomorphism, since any constant function satisfies the relation.
Now, with this observation, you can show the following, which makes the definition more clear:
Lemma Let $f:L\to M$ be any function. Then $f$ is order isomomorphism if and only if $f : L \to Im(f)$ is an order homomorphism and its inverse $f^{-1}: Im(f) \to L$ is an order homomorphism.
Note: In general in mathematics, the difference between 'if' and 'iff' is often huge, so I would not say that the difference is just "if" and "iff".