Shift Mapping is Lower Adjoint iff Appropriate Maxima Exist

Theorem
Let $\struct {S, \preceq}$ be a meet semilattice.

Then the following two conditions are equivalent:


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


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

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

Let $x, t \in S$.

Define $f: S \to S$:
 * $\forall s \in S: \map f s = x \wedge s$

By assumption:
 * $f$ is lower adjoint

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

By Galois Connection is Expressed by Maximum:
 * $\forall s \in S: \map g s = \map \max {f^{-1} \sqbrk {s^\preceq} }$

Then:
 * $\map \max {f^{-1} \sqbrk {t^\preceq} }$ exists.

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

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

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

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

As maxima exist define a mapping $g: S \to S$:
 * $\forall s \in S: \map g s = \map \max {f^{-1} \sqbrk {s^\preceq} }$

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

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

By definition of $f$:
 * $\map f y = x \wedge y$ and $\map f z = x \wedge z$

Thus by Meet Semilattice is Ordered Structure:
 * $\map f y \preceq \map f z$

By Galois Connection is Expressed by Maximum:
 * $\struct {g, f}$ is a Galois connection

By definition:
 * $f$ is a lower adjoint.