We give an axiomatic treatment of fixed-point operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the free iteration theory. We then show how iteration operators arise in axiomatic domain theory. One result derives them from the existence of sufficiently many bifree algebras (exploiting the universal property Freyd introduced in his notion of algebraic compactness) . Another result shows that, in the presence of a parameterized natural numbers object and an equational lifting monad, any uniform fixed-point operator is necessarily an iteration operator. 1. Introduction Fixed points play a central role in domain theory. Traditionally, one works with a category such as Cppo, the category of !-continuous functions between !-complete pointed partial orders. This possesses a least-fixed-point oper...

From the analogue of Böhm’s Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here of these results, which were obtained previously by Richard Statman and Alex K. Simpson.

We present the model construction technique called Linear Realizability. It consists in building a category of Partial Equivalence Relations over a Linear Combinatory Algebra. We illustrate how it can be used to provide models, which are fully complete for various typed λ-calculi. In particular, we focus on special Linear Combinatory Algebras of partial involutions, and we present PER models over them which are fully complete, inter alia, w.r.t. the following languages and theories: the fragment of System F consisting of ML-types, the maximal theory on the simply typed λ-calculus with finitely many ground constants, and the maximal theory on an infinitary version of this latter calculus. Key words: Typed lambda-calculi, ML-polymorphic types, linear logic, hyperdoctrines, PER models, Geometry of Interaction, (axiomatic) full completeness

In the Böhm theorem workshop on Crete island, Zoran Petric called Statman’s “Typical Ambiguity theorem ” typed Böhm theorem. Moreover, he gave a new proof of the theorem based on set-theoretical models of the simply typed lambda calculus. In this paper, we study the linear version of the typed Böhm theorem on a fragment of Intuitionistic Linear Logic. We show that in the multiplicative fragment of intuitionistic linear logic without the multiplicative unit 1 (for short IMLL) weak typed Böhm theorem holds. The system IMLL exactly corresponds to the linear lambda calculus without exponentials, additives and logical constants. The system IMLL also exactly corresponds to the free symmetric monoidal closed category without the unit object. As far as we know, our separation result is the first one with regard to these systems in a purely syntactical manner. 1

An arithmetical proof of the strong normalization

An arithmetical proof of the strong normalization for the λ-calculus with recursive equations on types