From lambda-Calculus to Universal Algebra and Back

We generalize to universal algebra concepts originating from lambda calculus and programming in order first to prove a new result on the lattice of λ-theories, and second a general theorem of pure universal algebra which can be seen as a meta version of the Stone Representation Theorem. The interest of a systematic study of the lattice λT of λ-theories grows out of several open problems on lambda calculus. For example, the failure of certain lattice identities in λT would imply that the problem of the orderincompleteness of lambda calculus raised by Selinger has a negative answer. In this paper we introduce the class of Church algebras (which includes all Boolean algebras, combinatory algebras, rings with unit and the term algebras of all λ-theories) to model the if-then-else instruction of programming and to extend some properties of Boolean algebras to general universal algebras. The interest of Church algebras is that each has a Boolean algebra of central elements, which play the role of the idempotent elements in rings. Central elements are the key tool to represent any Church algebra as a weak Boolean product of directly indecomposable Church algebras and to prove the meta representation theorem mentioned above. We generalize the notion of easy λ-term and prove that any Church algebra with an “easy set” of cardinality n admits (at the top) a lattice interval of congruences isomorphic to the free Boolean algebra with n generators. This theorem has the following consequence for λT : for every recursively enumerable λtheory φ and each n, there is a λ-theory φn ≥ φ such that {ψ : ψ ≥ φn} “is” the Boolean lattice with 2 elements.