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The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way Boolean algebras algebraize the classical propositional calculus. The equational theory of lambda abstraction algebras is intended as an alternative to combinatory logic in this regard since it is a ÿrst-order algebraic… (More)

A model of the untyped lambda calculus univocally induces a lambda theory (i.e., a congruence relation on λ-terms closed under α- and β-conversion) through the kernel congruence relation of the interpretation function. A semantics of lambda calculus is <i>(equationally) incomplete</i> if there exists a lambda theory that is not induced by… (More)

In this paper we show that the Stone representation theorem for Boolean algebras can be generalized to combi-natory 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… (More)

Scott's information systems provide a categorically equivalent, intensional description of Scott domains and continuous functions. Following a well established pattern in denotational semantics, we define a linear version of information systems, providing a model of intuitionistic linear logic (a new-Seely category), with a " set-theoretic " interpretation… (More)

Lambda abstraction algebras (LAAs) are designed to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order predicate logic. Like combinatory algebras they can be defined by true identities and thus form a variety in the sense of universal algebra, but they differ from combinatory algebras in several… (More)

Lambda theories are equational extensions of the untyped lambda calculus that are closed under derivation. The set of lambda theories is naturally equipped with a structure of complete lattice, where the meet of a family of lambda theories is their intersection, and the join is the least lambda theory containing their union. In this paper we study the… (More)

The notion of institution is dissected into somewhat weaker notions. We introduce a novel notion of institution morphism, and characterize preservation of institution properties by corresponding properties of such morphisms. Target of this work is the stepwise construction of a general framework for translating logics, and algebraic specifications using… (More)