Looking for example of an order homomorphism that doesn't preserve joins.

Question

I know that not every order homomorphism preserves joins. But, I can't think of an example!

Both minimal examples and 'natural' examples welcome.

Answer 1

Let $S$ be the poset

                                                                   

and let $T$ be the poset

                                                                   

Then the map $\phi:S\to T$ defined by $$\phi(a)=y,\quad \phi(b)=z,\quad\phi(c)=w$$ is an order homomorphism, but $$\phi(a\vee b)=\phi(c)=w\neq x=\phi(a)\vee\phi(b).$$