Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice

Theorem

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

Let $a, x, y \in S$.

Then

$a \wedge x \preceq y$ if and only if $x \preceq a \to y$

Proof

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

$\forall s \in S: \map d s = a \wedge s$

Define a mapping $g: S \to S$:

$\forall s \in S: \map g s = a \to s$

By Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice:

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

By definition of Galois connection:

$x \preceq \map g y$ if and only if $\map d x \preceq y$

Thus by definitions of $g$ and $d$:

$a \wedge x \preceq y$ if and only if $x \preceq a \to y$

$\blacksquare$

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_1:67