Representation of lattices via set-colored posets

This paper proposes a representation theory for any finite lattice via set-colored posets, in the spirit of Birkhoff for distributive lattices. The notion of colored posets was introduced in Nourine (2000) [34] and the generalization to set-colored posets was given in Nourine (2000) [35]. In this paper, we give a characterization of set-colored posets for general lattices, and show that set-colored posets capture the order induced by join-irreducible elements of a lattice as Birkhoff's representation does for distributive lattices. We also give a classification for some lattices according to the coloring property of their set-colored representation including upper locally distributive, upper locally distributive, meet-extremal and semidistributive lattices.