Abstract not found

In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the first-order case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. Some of the most basic results in classification theory for non elementary classes are presented. The motivating point of view is Shelah's categoricity conjecture for L# 1 ,# . While only very basic theorems are proved, an effort is made to present number of different technologies: Flavors of weak diamond, models of weak set theories, and commutative diagrams. We focus in issues involving existence of Galois types, extensions of types and Galois-stability.

this article is on protocols allowing the well-functioning parts of such a large and complex system to carry out their work despite the failure of others. Many deep and interesting results on such problems have been discovered by computer scientists in recent years, the incorporation of which into game theory can greatly enrich this field

We outline a simple approach to axiomatising the class of representable relation algebras, using games. We discuss generalisations of the method to cylindric algebras, homogeneous and complete representations, and atom structures of relation algebras. 1 Introduction Relation algebras are to binary relations what boolean algebras are to unary ones. They are used in artificial intelligence, where, for example, the Allen--Koomen temporal planning system checks the consistency of given relations between time intervals. In mathematics, they form a part of algebraic logic. The history of this goes back to the nineteenth century, the early workers including Boole, de Morgan, Peirce, and Schroder; it was studied intensively by Tarski's group (including, at various times, Chin, Givant, Henkin, J'onsson, Lyndon, Maddux, Monk, N'emeti) from around the 1950s, and currently we know of active groups in Amsterdam, Budapest, Rio de Janeiro, South Africa, and the U.S., among other places. Abstract...

this paper, developing the theory of forking for the category of existentially closed models of an arbitrary universal theory T , assuming a suitable notion of "simplicity". Under the additional assumption that T has the amalgamation property (AP) and joint embedding property (JEP), Shelah did develop forking in some form. Hrushovski ([4]) rediscovered this latter class of theories (universal T with AP and JEP) calling them Robinson theories, and pointing out that all model-theoretic methods should apply to the category of e.c. models of such T . For Robinson theories, quantifier-free types are the main object of study, and as quantifier-free formulas are closed # Supported by NSF grant DMS-96-96268

We present a general framework for carrying out the constructions in [2-10] and others of the same type. The unifying factor is a combinatorial principle which we present in terms of a game in which the first player challenges the second player to carry out constructions which would be much easier in a generic extension of the universe, and the second player cheats with the aid of .

Dov Gabbay is a prolific logician just by himself. But beyond that, he is quite good at making other people investigate the many further things he cares about. As a result, King's College London has become a powerful attractor in our field worldwide. Thus, it is a great pleasure to be an organizer for one of its flagship events: the Augustus de Morgan Workshop of 2005. Benedikt Loewe and I proposed the topic of 'interactive logic ' for this occasion, with an emphasis on social software – the logical analysis and design of social procedures – and on games, arguably the formal interactive setting par excellence. This choice reflects current research interests in our logic community at ILLC Amsterdam and beyond. In this broad area of interfaces between logic, computer science, and game theory, this paper is my own attempt at playing Dov. I am, perhaps not telling, but at least asking other people to find out for me what I myself cannot. A word of historical clarification may help here. The last time the Dutch came up the Thames (in 1667), we messed up the harbour, burnt down a few buildings, and took the English flagship the Royal Charles with us as a souvenir. The Medway Raid was still commemorated as late as 1967 in a joint ceremony. This time, however, our intentions

Abstract. The class of all Artinian local rings of length at most l is ∀2-elementary, axiomatised by a finite set of axioms Artl. We show that its existentially closed models are Gorenstein, of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory Gorl of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory Artl is companionable, with model-companion Gorl. In studying classes of commutative rings with identity (rings, for short) from a model-theoretic point of view, the elementary or (first order) axiomatisable classes are the most interesting. Given a general property of rings, one might look for elementary classes all of whose models share this property. Among their beloved properties, commutative algebraists will no doubt quote the Noetherian

this paper to explore this phenomenon, both its extent and its limits, in greater detail. As a first step, we introduce an analogue of the expansion of a language by Skolem functions. Consider a language L 82 which extends a first-order language with equality by introducing a unary generalized quantifier 2 x . We use this notation to emphasize an analogy between generalized quantifiers and modal operators, that will become apparent below. The dual of 2 x is 3 x ' = df :2 x :'. (In the examples above a filter quantifier would correspond to 2.)

iii In memory of my father. iv