In this paper we show that the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras. In every combinatory algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings), whose operations are defined by suitable combinators. Central elements are used to represent any combinatory algebra as a Boolean product of directly indecomposable combinatory algebras (i.e., algebras which cannot be decomposed as the Cartesian product of two other nontrivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e., the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible lambda theories. In one of the main results of the paper we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible: for every recursively enumerable lambda theory T, there is a continuum of lambda theories including T which are omitted by the indecomposable semantics. 1

Abstract. A longstanding open problem is whether there exists a nonsyntactical model of the untyped λ-calculus whose theory is exactly the least λ-theory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λ-calculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of λ-calculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ, λβη. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum one. Finally, we show that the class of graph models enjoys a kind of downwards Löwenheim-Skolem theorem.

A longstanding open problem in lambda calculus is whether there exists a non-syntactical model of the untyped lambda calculus whose theory is exactly the least λ-theory λβ. In this paper we make use of the Visser topology for investigating the related question of whether the equational theory of a model can be recursively enumerable (r.e. for brevity). We introduce the notion of an effective model of lambda calculus and prove the following results: (i) The equational theory of an effective model cannot be λβ, λβη; (ii) The order theory of an effective model cannot be r.e.; (iii) No effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate the class of graph models and prove the following, where “graph theory ” is a shortcut for “theory of a graph model”: (iv) There exists a minimum order graph theory (for equational graph theories this was proved in [9, 10]). (v) The minimum equational/order graph theory is the theory of an effective graph model. (vi) No order graph theory can be r.e. (vii) Every equational/order graph theory is the theory of a graph model having a countable web. This last result proves that the class of graph models enjoys a kind of (downwards) Löwenheim-Skolem theorem, and it answers positively Question 3 in [4, Section 6.3] for the class of graph models. 1.

Abstract. We generalize to universal algebra concepts originating from λ-calculus and programming to prove a new result on the lattice λT of λ-theories, and a general theorem of pure universal algebra which can be seen as a meta version of the Stone Representation Theorem. 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. 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 indecomposable Church algebras and to prove the meta representation theorem mentioned above. We generalize the notion of easy λ-term of lambda calculus to 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 the lattice of λ-theories: for every recursively enumerable λ-theory φ and each n, there is a λ-theory φn ⊇ φ such that {ψ: ψ ⊇ φn} “is ” the Boolean lattice with 2 n elements.

A longstanding open problem is whether there exists a model of the untyped λ-calculus in the category Cpo of complete partial orderings and Scott continuous functions, whose theory is exactly the least λ-theory λβ or the least extensional λ-theory λβη. In this paper we analyze the class of reflexive Scott domains, the models of λ-calculus living in the category of Scott domains (a full subcategory of Cpo). The following are the main results of the paper: (i) Extensional reflexive Scott domains are not complete for the λβη-calculus, i.e., there are equations not in λβη which hold in all extensional reflexive Scott domains. (ii) The order theory of an extensional reflexive Scott domain is never recursively enumerable. These results have been obtained by isolating among the reflexive Scott domains a class of webbed models arising from Scott’s information systems, called iweb-models. The class of iweb-models includes all extensional reflexive Scott domains, all preordered coherent models and all filter models living in Cpo. Based on a fine-grained study of an “effective” version of Scott’s information systems, we have shown that there are equations not in λβ (resp. λβη) which hold in all (extensional) iweb-models.

The aim of this paper is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λ-theories ( = equational extensions of untyped λ-calculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to several questions raised in these years as well as several independent results, the state of the art about the long-standing open questions concerning the representability of λ-theories as theories of models, and 26 open problems. On the other side, against the common belief, we show that lambda calculus and combinatory logic satisfy interesting algebraic properties. In fact the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras and λ-abstraction algebras. In every combinatory and λ-abstraction algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings). Central elements are used to represent any combinatory and λ-abstraction algebra as a weak Boolean product of directly indecomposable algebras (i.e., algebras which cannot be decomposed as the Cartesian product of two other non-trivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e., the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible λ-theories. In one of the main results of the paper we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible.

A longstanding open problem is whether there exists a non-syntactical model of the untyped λ-calculus whose theory is exactly the least λ-theory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λ-calculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of λ-calculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ, λβη. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum among the theories of graph models. Finally, we show that the class of graph models enjoys a kind of downwards Löwenheim-Skolem theorem.

Abstract. This paper is an exploration in a functional programming framework of isomorphisms between elementary data types (natural numbers, sets, multisets, finite functions, permutations binary decision diagrams, graphs, hypergraphs, parenthesis languages, dyadic rationals, primes, DNA sequences etc.) and their extension to hereditarily finite universes through hylomorphisms derived from ranking/unranking and pairing/unpairing operations. An embedded higher order combinator language provides any-to-any encodings automatically. Besides applications to experimental mathematics, a few examples of “free algorithms ” obtained by transferring operations between data types are shown. Other applications range from stream iterators on combinatorial objects to self-delimiting codes, succinct data representations and generation of random instances. The paper covers 60 data types and, through the use of the embedded combinator language, provides 3660 distinct bijective transformations between them. The self-contained source code of the paper, as generated from a literate Haskell program, is available at