Labeled posets are universal

Partially ordered sets labeled with kk labels (kk-posets) and their homomorphisms are examined. The homomorphicity order of finite kk-posets is shown to be a distributive lattice. Homomorphicity orders of finite kk-posets and kk-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 kk-posets and their digraph representations is established.