All Infima Preserving Mapping is Upper Adjoint of Galois Connection/Lemma 1

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Theorem

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

Let $\struct {T, \precsim}$ be an ordered set.

Let $g: S \to T$ be an all infima preserving mapping.


Then:

$g$ is an increasing mapping.


Proof

Define a mapping $d: T \to S$:

$\forall t \in T: \map d t := \map \inf {g^{-1} \sqbrk {t^\succsim} }$

Let $x, y \in S$ such that

$x \preceq y$

By Upper Closure is Decreasing:

$y^\succeq \subseteq x^\succeq$

By Infimum of Upper Closure of Element:

$\map \inf {x^\succeq} = x$ and $\map \inf {y^\succeq} = y$

By definition of all infima preserving mapping:

$g$ preserves the infimum on $x^\succeq$

and:

$g$ preserves the infimum on $y^\succeq$

By definition of infimum on subset preserving mapping:

$\map \inf {\map {g^\to} {x^\succeq} } = \map g x$ and $\map \inf {\map {g^\to} {y^\succeq} } = \map g y$

By Image of Subset under Mapping is Subset of Image:

$\map {g^\to} {y^\succeq} \subseteq \map {g^\to} {x^\succeq}$

Thus by Infimum of Subset:

$\map g x \precsim \map g y$

$\blacksquare$


Sources