Why aren't these lattices isomorphic?

[twoflower]

Hi,

I can't see why these lattices aren't isomorphic:

<br /> (\mathbb{N} \mbox{ u } \left\{-\infty, +\infty\right\}, \le) \mbox{ and } (\mathbb{N} \mbox{ u } \left\{-\infty, +\infty\right\}, \le)^{inverse}<br /> [/itex]<br /> <br /> I thought that this isomorphism would straightforwardly map an element <b>x</b> onto <b>-x</b> in the second lattice, so why aren&#039;t these isomorphic please?<br /> <br /> I see why <b>these</b> two lattices aren&#039;t isomorphic:<br /> <br /> &lt;br /&gt; (\mathbb{N}, \le) \mbox{ and } (\mathbb{N}, \le)^{inverse}&lt;br /&gt; [/itex]&lt;br /&gt; &lt;br /&gt; because the element &lt;b&gt;0&lt;/b&gt; from the first one has no equivalent in the inversed lattice, but the first couple of lattices seems ok to me.&lt;br /&gt; &lt;br /&gt; &lt;br /&gt; &lt;br /&gt; &lt;br /&gt; Thank you for clarification.

 

[StatusX]

N consists only of positive integers. If you meant Z, all integers, the first pair are isomorphic.