Initial Semantics for Strengthened Signatures

André Hirschowitz (1) , Marco Maggesi (2)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 Dipartimento di Matematica "Ulisse Dini"

We give a new general definition of arity, yielding the companion notions of signature and associated syntax. This setting is modular in the sense requested by [10]: merging two extensions of syntax cor- responds to building an amalgamated sum. These signatures are too general in the sense that we are not able to prove the existence of an as- sociated syntax in this general context. So we have to select arities and signatures for which there exists the desired initial monad. For this, we follow a track opened by Matthes and Uustalu [16]: we introduce a notion of strengthened arity and prove that the corresponding signa- tures have initial semantics (i.e. associated syntax). Our strengthened arities admit colimits, which allows the treatment of the λ-calculus with explicit substitution in the spirit of [10].

Mots clés: Syntax Semantics

Domaines: Mathématiques générales [math.GM]

André Hirschowitz, Marco Maggesi. Initial Semantics for Strengthened Signatures. 2012. ⟨hal-00762649⟩