# Homomorphisms between non-isomorphic finite lattices

Let $L_{1}$ and $L_{2}$ be non-isomorphic finite lattices of the same cardinality. Can there exist any lattice homomorphisms between $L_{1}$ and $L_{2}$?

• Yes, but you probably mean to ask a different question. There are constant maps, and in some cases maps to a two element interval. It depends (partly) on what congruences each lattice has. Gerhard "Maybe Your Lattices Are Special?" Paseman, 2018.08.04. – Gerhard Paseman Aug 4 '18 at 19:53
• @GerhardPaseman are constant maps lattice homomorphisms? – Tim Campion Aug 4 '18 at 19:54
• I assume lattice means algebraic structure, and he did not specify bounds. Yes the constant map preserves the algebraic relations, as the functions are idempotent. Gerhard "This Is What I Understand" Paseman, 2018.08.04. – Gerhard Paseman Aug 4 '18 at 20:00
• Ah, I was assuming the top and bottom elements should be preserved. Probably I'm mistaken about what the standard terminology is. – Tim Campion Aug 4 '18 at 20:00
• @TobiasFritz, thanks for the comment, but I'm not sure I get the implication. All finite Boolean algebras of the same cardinality are isomorphic, so I didn't think the Boolean cases would be pertinent here. What am I missing? – King Kong Aug 4 '18 at 20:58