Complete Lattice - Morphisms of Complete Lattices

Morphisms of Complete Lattices

The traditional morphisms between complete lattices are the complete homomorphisms (or complete lattice homomorphisms). These are characterized as functions that preserve all joins and all meets. Explicitly, this means that a function f: L→M between two complete lattices L and M is a complete homomorphism if

• and
• ,

for all subsets A of L. Such functions are automatically monotonic, but the condition of being a complete homomorphism is in fact much more specific. For this reason, it can be useful to consider weaker notions of morphisms, that are only required to preserve all meets or all joins, which are indeed inequivalent conditions. This notion may be considered as a homomorphism of complete meet-semilattices or complete join-semilattices, respectively.

Furthermore, morphisms that preserve all joins are equivalently characterized as the lower adjoint part of a unique Galois connection. Each of these determines a unique upper adjoint in the inverse direction that preserves all meets. Hence, considering complete lattices with complete semilattice morphisms boils down to considering Galois connections as morphisms. This also yields the insight that the introduced morphisms do basically describe just two different categories of complete lattices: one with complete homomorphisms and one with meet-preserving functions (upper adjoints), dual to the one with join-preserving mappings (lower adjoints).