Using Vaggione's concept of central element in a double-pointed algebra, we introduce the notion of Boolean-like variety as a generalisation of Boolean algebras to an arbitrary similarity type. Appropriately relaxing the requirement that every element be central in any member of the variety, we obtain the more general class of semi-Boolean-like varieties, which still retain many of the pleasing properties of Boolean algebras. We prove that a double-pointed variety is discriminator if and only if it is semi-Boolean-like, idempotent, and 0-regular. This theorem yields a new Maltsev-style characterisation of double-pointed discriminator varietie

A model of the untyped lambda calculus univocally induces a lambda theory (i.e., a congruence relation on λ-terms closed under αand β-conversion) through the kernel congruence relation of the interpretation function. A semantics of lambda calculus is (equationally) incomplete if there exists a lambda theory that is not induced by any model in the semantics. In this article, we introduce a new technique to prove in a uniform way the incompleteness of all denotational semantics of lambda calculus that have been proposed so far, including the strongly stable one, whose incompleteness had been conjectured by Bastonero, Gouy and Berline. We apply this technique to prove the incompleteness of any semantics of lambda calculus given in terms of partially ordered models with a bottom element. This incompleteness removes the belief that partial orderings with a bottom element are intrinsic to models of the lambda calculus, and that the incompleteness of a semantics is only due to the richness of the structure of representable functions. Instead, the incompleteness is also due to the richness of the structure of lambda theories. Further results of the article are: (i) an incompleteness theorem for partially ordered models with finitely many connected components (= minimal upward and downward closed sets); (ii) an incompleteness theorem for topological models whose topology satisfies a suitable property of connectedness; (iii) a completeness theorem for topological models whose topology is non-trivial and metrizable.

Equational type logic is an extension of (conditional) equational logic, that enables one to deal in a single, unified framework with diverse phenomena such as partiality, type polymorphism and dependent types. In this logic, terms may denote types as well as elements, and atomic formulae are either equations or type assignments. Models of this logic are type algebras, viz. universal algebras equipped with a binary relation-to support type assignment. Equational type logic has a sound and complete calculus, and initial models exist. The use of equational type logic is illustrated by means of simple examples, where all of the aforementioned phenomena occur. Formal notions of reduction and extension are introduced, and their relationship to free constructions is investigated. Computational aspects of equational type logic are investigated in the framework of conditional term rewriting systems, genera!izing known results on confluence of these systems. Finally, some closely related work is reviewed and future research directions are outlined in the conclusions. * 41, Ird.wy'~~-rted (conditional) equational logic is the most established basis to the algebraic approach to abstract data type (ADT) specification [ 15,9]. algebras [ 17,4] are the standard models of this lo&c, extending u structures in a straightforward way. In algebraic specification, however, several phenomena indicate that this logic encounters limitations in practice. We mention a few, most interesting of these phenomena (which are discussed in Section 2): partiaiity, exception handling, extension, type polymorphism, dependent types. Several formal frameworks have been designed to solve the problems that are raised by e&r& of these phenomena. In particular, many of these frameworks are based on extensions of equational logic in various forms. Most of these approaches address the phenomena of their interest at a rather high level of generality. Yet, a unifying approach, where all of these phenomena can be dealt with, does not seem to have emerged. The following problem is addressed in this paper: to nd and investigate a parsimonious logic of types where al2 of the aforementioned phenomena can be dealt with in an algebraic setting. In Section 2 we further motivate our irwstigation ng, for each of those phenomena, rt discussion an a simple exam@ational type logic

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 structure of the lattice of lambda theories by universal algebraic methods. We show that nontrivial quasi-identities in the language of lattices hold in the lattice of lambda theories, while every nontrivial lattice identity fails in the lattice of lambda theories if the language of lambda calculus is enriched by a suitable finite number of constants. We also show that there exists a sublattice of the lattice of lambda theories which satisfies: (i) a restricted form of distributivity, called meet semidistributivity; and (ii) a nontrivial identity in the language of lattices enriched by the relative product of binary relations.

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We generalize Baeten and Boerbom's method of forcing, and apply it to show that there is a fixed sequence (uk)k in N, of closed untyped λ-terms satisfying the following properties: * For any countable sequence (gk)k in N, of continuous functions (of arbitrary arity) on the power set P(D) of an arbitrary countable set D, there is a graph model (D,p) such that (λx.xx)(λx.xx)uk represents gk in the model. * For any countable sequence (tk)k in N, of closed λ-terms there is a graph model that satisfies (λx.xx)(λx.xx)uk=tk for all k. These two results bring information on the landscape of graph theories (= λ-theories that can be realized as theories of graph models), and more generally on the lattice of λ-theories (ordered by inclusion)

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[7, Section 6.2] for the restricted class of graph models).