A functorial semantics for multi-algebras and partial algebras, with applications to syntax

Multi-algebras allow for the modelling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multi-algebras and partial algebras, analogous to the classical presentation of algebras over a signature Sigma as cartesian functors from the algebraic theory over Sigma to Set. We introduce two different notions of theory over a signature, both having a structure weaker than cartesian, and we consider functors; from them to Rel or Pfn, the categories of sets and relations or partial functions, respectively. Next we discuss how the functorial presentation provides guidelines when choosing syntactical notions for a class of algebras, and as an application we argue that the natural generalization of usual terms are "conditioned terms" for partial algebras, and "term graphs" for multi-algebras. (C) 2002 Elsevier Science B.V. All rights reserved.