Shift Mapping is Lower Adjoint iff Appropriate Maxima Exist

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

Then the following two conditions are equivalent:

$(1): \quad \forall x \in S, f:S \to S: \left({\forall s \in S: f\left({s}\right) = x \wedge s}\right) \implies f$ is lower adjoint

$(2): \quad \forall x, t \in S: \max \left\{ {s \in S: x \wedge s \preceq t}\right\}$ exists.

$(1) \implies (2)$
Assume that
 * $\forall x \in S, f:S \to S: \left({\forall s \in S: f\left({s}\right) = x \wedge s}\right) \implies f$ is lower adjoint

Let $x, t \in S$.

Define $f: S \to S$:
 * $\forall s \in S: f\left({s}\right) = x \wedge s$

By assumption:
 * $f$ is lower adjoint

By definition of lower adjoint:
 * $\exists g: S \to S: \left({g, f}\right)$ is Galois connection

By Galois Connection is Expressed by Maximum:
 * $\forall s \in S: g\left({s}\right) = \max\left({f^{-1}\left[{s^\preceq}\right]}\right)$

Then
 * $\max\left({f^{-1}\left[{t^\preceq}\right]}\right)$ exists.

By definition of image of set:
 * $\max\left\{ {s \in S: f\left({s}\right) \in t^\preceq}\right\}$ exists.

By definition of lower closure of element:
 * $\max \left\{ {s \in S: x \wedge s \preceq t}\right\}$ exists.

$(2) \implies (1)$
Assume that
 * $\forall x, t \in S: \max \left\{ {s \in S: x \wedge s \preceq t}\right\}$ exists.

Let $x \in S, f: S \to S$ such that
 * $\forall s \in S: f\left({s}\right) = x \wedge s$

As maxima exist define a mapping $g: S \to S$:
 * $\forall s \in S: g\left({s}\right) = \max\left({f^{-1}\left[{s^\preceq}\right]}\right)$

We will prove that
 * $f$ is an increasing mapping

Let $y, z \in S$ such that
 * $y \preceq z$

By definition of $f$:
 * $f\left({y}\right) = x \wedge y$ and $f\left({z}\right) = x \wedge z$

Thus by Meet Semilattice is Ordered Structure:
 * $f\left({y}\right) \preceq f\left({z}\right)$

By Galois Connection is Expressed by Maximum:
 * $\left({g, f}\right)$ is Galois connection

By definition
 * $f$ is a lower adjoint