  A finite poset X carries a natural structure of a topological space. Fix a
field k, and denote by D(X) the bounded derived category of sheaves of finite
dimensional k-vector spaces over X. Two posets X and Y are said to be derived
equivalent if D(X) and D(Y) are equivalent as triangulated categories.
  We give explicit combinatorial properties of a poset which are invariant
under derived equivalence, among them are the number of points, the
Z-congruency class of the incidence matrix, and the Betti numbers.
  Then we construct, for any closed subset Y of X, a strongly exceptional
collection in D(X) and use it to show an equivalence between D(X) and the
bounded derived category of a finite dimensional algebra A (depending on Y). We
give conditions on X and Y under which A becomes an incidence algebra of a
poset.
  We deduce that a lexicographic sum of a collection of posets along a
bipartite graph is derived equivalent to the lexicographic sum of the same
collection along the opposite graph. This construction produces many new
derived equivalences of posets and generalizes other well known ones.
  As a corollary we show that the derived equivalence class of an ordinal sum
of two posets does not depend on the order of summands. We give an example that
this is not true for three summands.
