Reflexive combinatory algebras

TL;DR

This paper introduces strongly reflexive combinatory algebras, providing a finite axiomatization and characterizing lambda algebras as stable, reflexive structures, thus connecting combinatory models with lambda calculus.

Contribution

It defines strongly reflexive combinatory algebras, offers a finite axiomatisation, and characterizes lambda algebras within this framework, linking models and lambda calculus.

Findings

Strongly reflexive combinatory algebras admit a finite axiomatization.

Lambda algebras are characterized as stable, strongly reflexive combinatory algebras.

A canonical construction relates strongly reflexive combinatory algebras to lambda algebras.

Abstract

We introduce the notion of reflexivity for combinatory algebras. Reflexivity can be thought of as an equational counterpart of the Meyer-Scott axiom of combinatory models, which indeed allows us to characterise an equationally definable counterpart of combinatory models. This new structure, called strongly reflexive combinatory algebra, admits a finite axiomatisation with seven closed equations, and the structure is shown to be exactly the retract of combinatory models. Lambda algebras can be characterised as strongly reflexive combinatory algebras which are stable. Moreover, there is a canonical construction of a lambda algebra from a strongly reflexive combinatory algebra. The resulting axiomatisation of lambda algebras by the seven axioms for strong reflexivity together with those for stability is shown to correspond to the axiomatisation of lambda algebras due to Selinger [J.Funct.Programming, 12(6), 549--566, 2002].