Labeled posets are universal

European J. Combin., 29 (2008), 493-506
http://dx.doi.org/10.1016/j.ejc.2007.02.005

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.