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*<math>\mathrm{bind}: (E \rarr \mathrm{M} \, A) \rarr (A \rarr E \rarr \mathrm{M}\,B) \rarr E \rarr \mathrm{M}\,B = m \mapsto k \mapsto e \mapsto \mathrm{bind} \, (m \, e) \,( a \mapsto k \, a \, e)</math>
*<math>\mathrm{lift}: \mathrm{M} \, A \rarr E \rarr \mathrm{M} \, A = a \mapsto e \mapsto a</math>
=== The state monad transformer ===
Given any monad <math>\mathrm{M} \, A</math>, the state monad transformer <math>S \rarr \mathrm{M}(A \times S)</math> (where <math>S</math> is the state type) is defined by:
*<math>\mathrm{return}: A \rarr S \rarr \mathrm{M} (A \times S) = a \mapsto s \mapsto \mathrm{return} \, (a, s)</math>
*<math>\mathrm{bind}: (S \rarr \mathrm{M}(A \times S)) \rarr (A \rarr S \rarr \mathrm{M}(B \times S)) \rarr S \rarr \mathrm{M}(B \times S) = m \mapsto k \mapsto s \mapsto \mathrm{bind} \, (m \, s) \,((a, s') \mapsto k \, a \, s')</math>
*<math>\mathrm{lift}: \mathrm{M} \, A \rarr S \rarr \mathrm{M}(A \times S) = a \mapsto s \mapsto \mathrm{bind} \, m \, (a \mapsto \mathrm{return} \, (a, s))</math>
== See also ==
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