User:Leigh.Samphier/Topology/Frame Homomorphism is Lower Adjoint of Unique Galois Connection

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Theorem

Let $L_1 = \struct{S_1, \preceq_1}, L_2 = \struct{L_2, \preceq_2}$ be frames.

Let $f : L_1 \to L_2$ be a frame homomorphism.

Then there exists a unique Galois connection $g = \tuple {g_*, g^*}$:

the lower adjoint $g^* = f$

Proof

By definition of frame homomorphism:

$f$ is arbitrary join preserving

From All Suprema Preserving Mapping is Lower Adjoint of Galois Connection:

there exists a Galois connection $g = \tuple {g_*, g^*} : g^* = f$

Furthermore, $g$ is necessarily unique.

$\blacksquare$