Definition of Dual Statement

Question

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.

Answer 1

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

Answer 2

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 $\langle X, \preceq\rangle$).