Congruence representations of join-homomorphisms of distributive lattices: a short proof
George Grätzer, H. Lakser, E. T. Schmidt · Czech digital mathematics library · 1996
Andras Huhn proved the following theorem: LetD andE be nite distributive lattices, and let :D!Ebe a f0g-preserving join-homomorphism. Then there are nite latticesK andLand there is a lattice homomorphism ':K!L such that ConK (the congruence lattice ofK) representsD, ConL (the congruence lattice of L) represents E, and the mapping ext': ConK! ConL (obtained by mapping a congruence ofK under'to L as a binary relation and then forming the minimal extension of this binary relation to a congruence relation of L) represents. In this note we give a short proof of this theorem. In fact, we prove amuch stronger result: forK one can choose any finite lattice whose congruence lattice is isomorphic to D.