Editing Heyting algebra (section)

===Lattice-theoretic definitions===
An equivalent definition of Heyting algebras can be given by considering the mappings:

:<math>\begin{cases} f_a \colon H \to H \\ f_a(x)=a\wedge x \end{cases}</math>

for some fixed ''a'' in ''H''. A bounded lattice ''H'' is a Heyting algebra [[if and only if]] every mapping ''f<sub>a</sub>'' is the lower adjoint of a monotone [[Galois connection]]. In this case the respective upper adjoint ''g<sub>a</sub>'' is given by ''g<sub>a</sub>''(''x'') = ''a''→''x'', where → is defined as above.

Yet another definition is as a [[residuated lattice]] whose monoid operation is ∧. The monoid unit must then be the top element 1. Commutativity of this monoid implies that the two residuals coincide as ''a''→''b''.