Semantics for algebraic operations

Abstract

Given a category C with finite products and a strong monad T on C, we investigate axioms under which an ObC-indexed family of operations of the form α_x : (Tx)n ! Tx provides a definitive semantics for algebraic operations added to the computational λ-calculus. We recall a definition for which we have elsewhere given adequacy results for both big and small step operational semantics, and we show that it is equivalent to a range of other possible natural definitions of algebraic operation. We outline examples and non-examples and we show that our definition is equivalent to one for call-by-name languages with effects too.