Relative Pseudocomplement Preserves Order

Theorem

Let $\struct {S, \vee, \wedge, \preceq}$ be a Brouwerian lattice.

Let $a, b, c \in S$.

Let $b \preceq c$.

Then

$a \to b \preceq a \to c$

where $x \to y$ denotes the relative pseudocomplement of $x$ with respect to $y$.

Proof

We have:

\(\ds a \to b\)

\(=\)

\(\ds \max \set {a \wedge x : x \in S : a \wedge x \preceq b}\)

Definition of Relative Pseudocomplement

\(\ds \)

\(\preceq\)

\(\ds \max \set {a \wedge x : x \in S : a \wedge x \preceq c}\)

Finer Supremum Precedes Supremum

\(\ds \)

\(=\)

\(\ds a \to c\)

Definition of Relative Pseudocomplement

$\blacksquare$