Algebraic duality for partially ordered sets

For an arbitrary partially ordered set P its dual P ∗ is built as the collection of all monotone mappings P → 2 where 2 = {0, 1} with 0 < 1. The set of mappings P ∗ is proved to be a complete lattice with respect to the pointwise partial order. The second dual P ∗∗ is built as the collection of all morphisms of complete lattices P ∗ → 2 preserving universal bounds. Then it is proved that the partially ordered sets P and P ∗∗ are isomorphic. AMS classification: 06A06, 06A15 Introduction The results presented in this paper can be considered as the algebraic counterpart of the duality in the theory of linear spaces. The outline of the construction looks as follows. Several categories occur in the theory of partially ordered sets. The most general is the category POSET whose objects are partially ordered sets and the morphisms are the monotone mappings. Another category which will be used is BCL whose objects are (bounded) complete lattices and the morhisms are the lattice homomorphisms preserving universal bounds. Evidently BCL is the subcategory of POSET .