This paper gives some sufficient conditions for admissible rules to be derivable in intuitionistic propositional calculus. For example, if the premises are Harrop formulas, the rule is admissible only if it is derivable.

In deriving the results, a particular class of substitutes is introduced, which are also useful when dealing with other questions of admissibility.

In the theory of denotational semantics of programming languages, several authors have constructed various kinds of universal domains. We present here a categorical generalization of a well-known result in model theory, which we use to characterize large classes of reasonable categories that contain universal homogeneous objects. The existence of such objects is characterized by the condition that the finite objects in the category satisfy the amalgamation property. We derive from this the existence and uniqueness of universal homogeneous domains for several categories of bifinite domains, with embedding-projection-pairs as morphisms. We also obtain universal homogeneous objects for various categories of stable bifinite domains. In contrast, several categories of event domains and concrete domains and the category of all coherent Scott-domains do not contain universal homogeneous objects. Finally, we show that all our constructions can be performed effectively.

Various notions of observing and testing processes are placed in a uniform algebraic framework in which observations are taken as constituting a quantale. General completeness criteria are stated, and proved in our applications.

Least fixpoints are constructed for finite coproducts of definable endofunctors of Cartesian closed categories that have weak polynomial products and joint equalizers of arbitrary families of pairs of parallel arrows. Both conditions hold in PER, the category whose objects are partial equivalence relations on N, and whose arrows are partial recursive functions. Weak polynomial products exist in any cartesian closed category with a finite number of objects as well as in any model of second order polymorphic lambda calculus: that is, in the proof theory of any second order positive intuitionistic propositional calculus, but such a category need not have equalizers. However, any finite coproduct of definable endofunctors of a cartesian closed category with weak polynomial products will have a least fixpoint in a larger category with equalizers whose objects are right ideals (or sieves) of modulo certain congruence relations, and whose arrows are induced from .

This paper introduces the following new constructions on stable domains and event structures: the tensor product; the linear function space; and the exponential. These give rise to a monoidal closed category of dI-domains and to stable event structures, which can be used to interpret intuitionistic linear logic. Finally, the usefulness of the category of stable event structures for modeling concurrency and its relation to other models are discussed.