Definition of Dual Statement In Order Theory, what is the exact definition of a dual statement? And what is the duality principle for posets/lattices? I haven't been able to find an exact statement or definition in this regard.
 A: Here's an example that might help you see what Hagen meant by "translates" in his answer.
Let $X = \{a,b,c\}$.  Let $\leq$ denote the binary relation $\{(a,a), (a,b), (b,b), (c,b), (c,c)\}$.  Then $\langle X, \leq \rangle$ is a poset, and the nontrivial pairs in the $\leq$ relation are $a\leq b$ and $c\leq b$.
If $\preceq$ denotes the binary relation $\{(a,a), (b,a), (b,b), (b,c), (c,c)\}$, then the nontrivial pairs are $b\preceq a$ and $b\preceq c$.  The relation $\preceq$ is dual to the relation $\leq$, and the poset $\langle X, \preceq\rangle$ is the dual of $\langle X, \leq \rangle$.
Here's a picture of these two posets.
The dual of the statement 
"$b$ is the largest element of $\langle X, \leq \rangle$" 
is the statement 
"$b$ is the smallest element of $\langle X, \preceq\rangle$".
The first statement (about $\langle X, \leq \rangle$) can be "translated" into a dual statement (about $\langle X, \preceq\rangle$).
A: If $(X,\le)$ is a poset, then so is $(X,\preceq)$ with $a\preceq b\iff b\le a$. Any statement about the former translates into a statement of the latter
