Brouwerian Lattice iff Shift Mapping is Lower Adjoint

Theorem
Let $\left({S, \preceq}\right)$ be a lattice.

Then $\left({S, \preceq}\right)$ is a Brouwerian lattice
 * $\forall x \in S, f: S \to S: \left({\forall s \in S: f \left({s}\right) = x \wedge s}\right) \implies f$ is a lower adjoint

Proof

 * $\left({S, \preceq}\right)$ is a Brouwerian lattice


 * $\forall x, y \in S: x$ has relative pseudocomplement with respect to $y$ by definition of Brouwerian lattice


 * $\forall x, y \in S: \max \left\{ {s \in S: x \wedge s \preceq y}\right\}$ exists by definition of relative pseudocomplement


 * $\forall x \in S, f: S \to S: \left({\forall s \in S: f \left({s}\right) = x \wedge s}\right) \implies f$ is a lower adjoint by Shift Mapping is Lower Adjoint iff Appropriate Maxima Exist