Elgot Algebras

TL;DR

This paper introduces Elgot algebras, a new algebraic structure for denotational semantics that ensures unique solutions to recursive specifications, grounded in iterative theories and axioms like functoriality and compositionality.

Contribution

It proposes Elgot algebras as a practical framework for semantics, connecting them to iterative theories and showing their categorical characterization.

Findings

Elgot algebras provide unique solutions for flat recursive equations.

The category of Elgot algebras is equivalent to the Eilenberg-Moore category of a specific monad.

The axioms of functoriality and compositionality are fundamental to the structure.

Abstract

Denotational semantics can be based on algebras with additional structure (order, metric, etc.) which makes it possible to interpret recursive specifications. It was the idea of Elgot to base denotational semantics on iterative theories instead, i.e., theories in which abstract recursive specifications are required to have unique solutions. Later Bloom and Esik studied iteration theories and iteration algebras in which a specified solution has to obey certain axioms. We propose so-called Elgot algebras as a convenient structure for semantics in the present paper. An Elgot algebra is an algebra with a specified solution for every system of flat recursive equations. That specification satisfies two simple and well motivated axioms: functoriality (stating that solutions are stable under renaming of recursion variables) and compositionality (stating how to perform simultaneous recursion). These two axioms stem canonically from Elgot's iterative theories: We prove that the category of Elgot algebras is the Eilenberg-Moore category of the monad given by a free iterative theory.