by So Kwok Yu, Andy. / Bibliography: leaves 64-65 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1985

The arithmetic progressions under addition and composition satisfy the usual rules of arithmetic with a modified distributive law. The basic algebra of such mathematical structures is examined; this leads to the consideration of the integers as a metric space under the "factoradic metric", i.e., the integers equipped with a distance function defined by d(n,m)=1/N!, where N is the largest positive integer such that N! divides n-m. Via the process of metric completion, the integers are then extended to a larger set of numbers, the factoradic integers. The properties of the factoradic integers are developed in detail, with particular attention to prime factorization, exponentiation, infinite series, and continuous functions, as well as to polynomials and their extensions. The structure of the factoradic integers is highly dependent upon the distribution of the prime numbers and relates to various topics in algebra, number theory, and non-standard analysis. / Master of Science

The main purpose of this thesis is to combine the categorical approach to logic given by the study of doctrines, with the universal algebraic techniques given by the theory of the pseudo-monads and pseudo-distributive laws. Every completions of doctrines is then formalized by a pseudo-monad, and then combinations of these are studied by the analysis of the pseudo-distributive laws. The starting point are the works of Maietti and Rosolini, in which they describe three completions for elementary doctrines: the first which adds full comprehensions, the second comprehensive diagonals, and the third quotients. Then we determine the existential completion of a primary doctrine, and we prove that the 2-monad obtained from it is lax-idempotent, and that the 2-category of existential doctrines is isomorphic to the 2-category of algebras for this 2-monad. We also show that the existential completion of an elementary doctrine is again elementary and we extend the notion of exact completion of an elementary existential doctrine to an arbitrary elementary doctrine. Finally we present the elementary completion for a primary doctrine whose base category has finite limits. In particular we prove that, using a general results about unification for first order languages, we can easily add finite limits to a syntactic category, and then apply the elementary completion for syntactic doctrines. We conclude with a complete description of elementary completion for primary doctrine whose base category is the free product completion of a discrete category, and we show that the 2-monad constructed from the 2-adjunction is lax-idempotent.

Dans cette thèse, on présente la théorie algébrique IH par le biais de générateurs et d’équations.Le modèle libre de IH est la catégorie des sous-espaces linéaires sur un corps k. Les termes de IH sont des diagrammes de cordes, qui, selon le choix de k, peuvent exprimer différents types de réseaux et de formalismes graphiques, que l’on retrouve dans des domaines scientifiques divers, tels que les circuits quantiques, les circuits électriques et les réseaux de Petri. Les équations de IH sont obtenues via des lois distributives entre algèbres de Hopf – d’où le nom “Interacting Hopf algebras” (algèbres de Hopf interagissantes). La caractérisation via les sous-espaces permet de voir IH comme une syntaxe fondée sur les diagrammes de cordes pour l’algèbre linéaire: les applications linéaires, les espaces et leurs transformations ont chacun leur représentation fidèle dans le langage graphique. Cela aboutit à un point de vue alternatif, souvent fructueux, sur le domaine.On illustre cela en particulier en utilisant IH pour axiomatiser la sémantique formelle de circuits de calculs de signaux, pour lesquels on s’intéresse aux questions de la complète adéquation et de la réalisabilité. Notre analyse suggère un certain nombre d’enseignements au sujet du rôle de la causalité dans la sémantique des systèmes de calcul. / We present by generators and equations the algebraic theory IH whose free model is the category oflinear subspaces over a field k. Terms of IH are string diagrams which, for different choices of k, expressdifferent kinds of networks and graphical formalisms used by scientists in various fields, such as quantumcircuits, electrical circuits and Petri nets. The equations of IH arise by distributive laws between Hopfalgebras - from which the name interacting Hopf algebras. The characterisation in terms of subspacesallows to think of IH as a string diagrammatic syntax for linear algebra: linear maps, spaces and theirtransformations are all faithfully represented in the graphical language, resulting in an alternative, ofteninsightful perspective on the subject matter. As main application, we use IH to axiomatise a formalsemantics of signal processing circuits, for which we study full abstraction and realisability. Our analysissuggests a reflection about the role of causality in the semantics of computing devices.