Galois Connection Implies Order on Mappings

Theorem

Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets.

Let $g: S \to T$ and $d: T \to S$ be mappings such that

$\tuple {g, d}$ is Galois connection.

Then $d \circ g \preceq I_S$ and $I_T \precsim g \circ d$

where

$\preceq, \precsim$ denote the orderings on mappings,

$I_S$ denotes the identity mapping of $S$

Let $s \in S$.

By definition of reflexivity:

$\map g s \precsim \map g s$

By definition of Galois connection:

$\map d {\map g s} \preceq s$

By definition of composition:

$\map {\paren {d \circ g} } s \preceq s$

By definition of identity mapping:

$\map {\paren {d \circ g} } s \preceq \map {I_S} s$

Thus by definition of order on mappings:

$d \circ g \preceq I_S$

Let $t \in T$.

By definition of reflexivity:

$\map d t \preceq \map d t$

By definition of Galois connection:

$t \precsim \map g {\map d t}$

By definition of composition:

$t \precsim \map {\paren {g \circ d} } t$

By definition of identity mapping:

$\map {I_T} t \precsim \map {\paren {g \circ d} } t$

Thus by definition of order on mappings:

$I_T \precsim g \circ d$

$\blacksquare$