  Earlier an arbitrary poset $P$ was proved to be isomorphic to the collection
of subsets of a space $M$ with two closures which are closed in the first
closure and open in the other. As a space $M$ for this representation an
algebraic dual space $P^*$ was used. Here we extend the theory of algabraic
duality for posets generalizing the notion of an ideal. This approach yields a
sufficient condition for the collection of clopen subsets of a subset of $P^*$
(with respect to induced closures) to be isomorphic to $P$. Applying this
result to certain classes of posets we prove some representation theorems and
get a topological characterization of orthocomplementations.
