Editing Complete lattice (section)

=== Galois connections and adjoints ===

Furthermore, morphisms that preserve all joins are equivalently characterized as the ''lower adjoint'' part of a unique [[Galois connection]]. For any pair of preorders ''X'' and ''Y'', a Galois connection is given by a pair of monotone functions ''f'' and ''g'' from ''X'' to ''Y'' such that for each pair of elements ''x'' of ''X'' and ''y'' of ''Y''

:<math>f(x) \leq y \iff x \leq g(y)</math>

where ''f'' is called the ''lower adjoint'' and ''g'' is called the ''upper adjoint''. By the [[adjoint functor theorem]], a monotone map between any pair of preorders preserves all joins if and only if it is a lower adjoint and preserves all meets if and only if it is an upper adjoint.

As such, each join-preserving morphism determines a unique ''upper adjoint'' in the inverse direction that preserves all meets. Hence, considering complete lattices with complete semilattice morphisms (of either type, join-preserving or meet-preserving) boils down to considering Galois connections as one's lattice morphisms. This also yields the insight that three classes of morphisms discussed above basically describe just two different categories of complete lattices: one with complete homomorphisms and one with Galois connections that captures both the meet-preserving functions (upper adjoints) and their [[duality (category theory)|dual]] join-preserving mappings (lower adjoints).

A particularly important class of special cases arises between lattices of subsets of ''X'' and ''Y'', i.e., the power sets {{tmath|P(X)}} and {{tmath|P(Y)}}, given a function {{tmath|f}} from ''X'' to ''Y''. In these cases, the direct image and inverse image maps induced by {{tmath|f}} between the power sets are upper and lower adjoints to each other, respectively.