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

Both minimal examples and 'natural' examples welcome.

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Let $S$ be the poset

                                                                   enter image description here

and let $T$ be the poset

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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 $$\phi(a\vee b)=\phi(c)=w\neq x=\phi(a)\vee\phi(b).$$

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  • $\begingroup$ Very minimal, thank you! $\endgroup$ – goblin May 17 '13 at 14:24
  • $\begingroup$ Try dollar sign \hspace{7cm} dollar sign. For generating empty space to move stuff around. $\endgroup$ – Nikolaj-K May 18 '13 at 23:49

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