Editing Heyting algebra (section)

===Definition===
Given two Heyting algebras ''H''<sub>1</sub> and ''H''<sub>2</sub> and a mapping {{nowrap|1=''f'' : ''H''<sub>1</sub> → ''H''<sub>2</sub>,}} we say that ''ƒ'' is a '''[[morphism]]''' of Heyting algebras if, for any elements ''x'' and ''y'' in ''H''<sub>1</sub>, we have:
#<math>f(0) = 0,</math>
#<math>f(x \land y) = f(x) \land f(y),</math>
#<math>f(x \lor y) = f(x) \lor f(y),</math>
#<math>f(x \to y) = f(x) \to f(y),</math>

It follows from any of the last three conditions (2, 3, or 4) that ''f'' is an increasing function, that is, that {{nowrap|1=''f''(''x'') ≤ ''f''(''y'')}} whenever {{nowrap|1=''x'' ≤ ''y''}}.

Assume ''H''<sub>1</sub> and ''H''<sub>2</sub> are structures with operations →, ∧, ∨ (and possibly ¬) and constants 0 and 1, and ''f'' is a surjective mapping from ''H''<sub>1</sub> to ''H''<sub>2</sub> with properties 1 through 4 above. Then if ''H''<sub>1</sub> is a Heyting algebra, so too is ''H''<sub>2</sub>. This follows from the characterization of Heyting algebras as bounded lattices (thought of as algebraic structures rather than partially ordered sets) with an operation → satisfying certain identities.