Shift Mapping is Lower Adjoint iff Meet is Distributive with Supremum

Theorem
Let $\struct {S, \preceq}$ be a complete lattice.

Then:
 * $\forall x \in S, f: S \to S: \paren {\forall y \in S: \map f y = x \wedge y} \implies f$ is lower adjoint of a Galois connection


 * $\forall x \in S, X \subseteq S: x \wedge \sup X = \sup \set {x \wedge y: y \in X}$
 * $\forall x \in S, X \subseteq S: x \wedge \sup X = \sup \set {x \wedge y: y \in X}$

Sufficient Condition
Assume that
 * $\forall x \in S, f: S \to S: \paren {\forall y \in S: \map f y = x \wedge y} \implies f$ is lower adjoint of a Galois connection

Let $x \in S, X \subseteq S$.

Define a mapping $f: S \to S$:
 * $\forall y \in S: \map f y := x \wedge y$

By assumption:
 * $f$ is lower adjoint of a Galois connection.

By Lower Adjoint Preserves All Suprema:
 * $f$ preserves all suprema.

By definition of mapping preserves all suprema
 * $f$ preserves the supremum of $X$.

By definition of complete lattice:
 * $X$ admits a supremum.

Thus

Necessary Condition
Assume that
 * $\forall x \in S, X \subseteq S: x \wedge \sup X = \sup \set {x \wedge y: y \in X}$

Let $x \in S$, $f: S \to S$ such that
 * $\forall y \in S: \map f y = x \wedge y$

We will prove that
 * $f$ preserves all suprema.

Let $X \subseteq S$ such that
 * $X$ admits a supremum.

Thus by definition of complete lattice:
 * $\map {f^\to} X$ admits a supremum.

Thus

Thus by definition
 * $f$ preserves the supremum of $X$.

Thus by All Suprema Preserving Mapping is Lower Adjoint of Galois Connection:
 * $f$ is lower adjoint of a Galois connection.