Editing Heyting algebra (section)

===Distributivity===
Heyting algebras are always [[distributivity (order theory)|distributive]]. Specifically, we always have the identities
#<math>a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)</math>
#<math>a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)</math>

The distributive law is sometimes stated as an axiom, but in fact it follows from the existence of relative pseudo-complements. The reason is that, being the lower adjoint of a [[Galois connection]], <math>\wedge</math> preserves all existing [[suprema]]. Distributivity in turn is just the preservation of binary suprema by <math>\wedge</math>.

By a similar argument, the following [[infinite distributive law]] holds in any complete Heyting algebra:

:<math>x\wedge\bigvee Y = \bigvee \{x\wedge y \mid y \in Y\}</math>

for any element ''x'' in ''H'' and any subset ''Y'' of ''H''. Conversely, any complete lattice satisfying the above infinite distributive law is a complete Heyting algebra, with
:<math>a\to b=\bigvee\{c\mid a\land c\le b\}</math>
being its relative pseudo-complement operation.