Labeled posets are universal

Partially ordered sets labeled with k labels ( k -posets) and their homomorphisms are examined. The homomorphicity order of finite k -posets is shown to be a distributive lattice. Homomorphicity orders of finite k -posets and k -lattices are shown to be universal in the sense that every countable poset can be embedded into them. Labeled posets are represented by directed graphs, and a categorical isomorphism between k -posets and their digraph representations is established.