Labeled posets are universal (Q2472843)

The main goal of this paper is to study partially ordered sets labeled with \( k\) labels (\(k\)-posets) and their homomorphisms. Section 2 deals with some preliminary results on these notions. In Section 3 it is proved that the homomorphicity order of finite \(k\)-posets is a distributive lattice. In Section 4 it is shown that the homomorphicity orders of finite \(k\)-posets and \(k\)-lattices are universal, in the sense that every countable poset can be embedded into them. Section 5 deals with representations of \(k\)-posets by directed graphs. An interesting categorical isomorphism between \(k\)-posets and their digraph representations is established.