All Infima Preserving Mapping is Upper Adjoint of Galois Connection

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 there exists a mapping $d: T \to S$ such that $\struct {g, d}$ is Galois connection and:

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

where:

$\min$ denotes the minimum

$g^{-1} \sqbrk {t^\succsim}$ denotes the image of $t^\succsim$ under relation $g^{-1}$

$t^\succsim$ denotes the upper closure of $t$

Proof

Lemma 1

$g$ is an increasing mapping.

$\Box$

Let us define a mapping $d: T \to S$ as:

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

Lemma 2

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

$\Box$

Thus by Galois Connection is Expressed by Minimum:

$\struct {g, d}$ is a Galois connection.

Thus by lemma $2$:

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

$\blacksquare$

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_1:14