Outline of a Calculus of Type Subsumption

This paper is a brief analysis of the notion of syntactic representation of types followed by a proposal of a formal calculus of type subsumption. The idea which is developed centers on the concept of indexed term, an extension of the definition of algebraic terms relaxing the fixed arity and fixed indexing constraints, and which allows term symbols to have some pre-order structure. It is shown that the structure on the set of symbols can be "homomorphically" extended to indexed terms to what is defined to be a subsumption ordering. Furthermore, when symbols have a lattice structure, this structure extends to a lattice of indexed terms. The notions of unification and generalization are also shown to fit the extension, and constitute the meet and join operations.