  In this work, we study 'reduction monads', which are essentially the same as
monads relative to the free functor from sets into multigraphs. Reduction
monads account for two aspects of the lambda calculus: on the one hand, in the
monadic viewpoint, the lambda calculus is an object equipped with a
well-behaved substitution; on the other hand, in the graphical viewpoint, it is
an oriented multigraph whose vertices are terms and whose edges witness the
reductions between two terms.
  We study presentations of reduction monads. To this end, we propose a notion
of 'reduction signature'. As usual, such a signature plays the role of a
virtual presentation, and specifies arities for generating
operations---possibly subject to equations---together with arities for
generating reduction rules. For each such signature, we define a category of
models; any model is, in particular, a reduction monad. If the initial object
of this category of models exists, we call it the 'reduction monad presented
(or specified) by the given reduction signature'.
  Our main result identifies a class of reduction signatures which specify a
reduction monad in the above sense. We show in the examples that our approach
covers several standard variants of the lambda calculus.
