Abstract. Inspired by the classical theory of modules over a monoid, we give a first account of the natural notion of module over a monad. The associated notion of morphism of left modules ("linear" natural transformations) captures an important property of compatibility with substitution, in the heterogeneous case where "terms" and variables therein could be of different types as well as in the homogeneous case. In this paper, we present basic constructions of modules and we show examples concerning in particular abstract syntax and lambda-calculus.

This paper is a contribution to the search for efficient and high-level mathematical tools to specify and reason about (abstract) programming languages or calculi. Generalising the reduction monads of Ahrens et al., we introduce transition monads, thus covering new applications such as-calculus,-calculus, Positive GSOS specifications, differential-calculus, and the big-step, simply-typed, call-by-value-calculus. Moreover, we design a suitable notion of signature for transition monads.

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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 operationspossibly 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.

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