Topological Representations of Posets

Details Topological Representations of Posets Topological Representations of Posets

2000-01-26

arxiv.org/abs/math/0001148v2

Mathematics

General Topology

7 pages, LaTeX 2e

Scientific paper

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.

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