Algebraic duality for partially ordered sets

Abstract

For an arbitrary partially ordered set $P$ its {\em dual} $P^{\ast }$ is built as the collection of all monotone mappings $P\to\2$ where $\2=\{0,1\}$ with $0<1$. The set of mappings $P^{\ast }$ is proved to be a complete lattice with respect to the pointwise partial order. The {\em second dual} $P^{\ast \ast }$ is built as the collection of all morphisms of complete lattices $P^*\to\2$ preserving universal bounds. Then it is proved that the partially ordered sets $P$ and $P^{\ast \ast }$ are isomorphic.

latex209, 6 pages