About Typed Algebraic Lambda-calculi

Arrighi and Dowek (2008) introduce an untyped lambdacalculus together with a structure of module over a ring of scalars. They show that diverging terms may break confluence if a naive rewriting system is used. In this paper, we explore the semantics of a typed version of their language. First, we describe a simply-typed language with no fixpoints. We show that in this restricted setting, no diverging term exists, and the language admits a simple denotation using the monad coming from the adjunction between the category of sets and the category of modules. Then we add the possibility of fixpoints, and we examine the differences with the previous language. In particular, we show how several notions of zeros and infinities naturally arise in the presence of fixpoints and we sketch a denotational description for catching these notions.