We will study some aspects of the local structure of models of certain C-minimal theories. We will prove (theorem 19) that, in a sufficiently saturated C-minimal structure in which the algebraic closure has the exchange property and which is locally modular, we can construct an infinite type-definable group around any non trivial point (a term to be defined later).

In most published systems of modal logic only one or two modal operators occur as undefined constants alongside the primitives of non-modal logic. But in a single informal argument many non-interdefinable modal operators may occur. E.g. in philosophy we may have ‘it is logically true that’, ‘it is analytic that’, and ‘it is physically necessary that’; and in jurisprudence or private international law we may have statements about what it is obligatory or not obligatory to do under several different systems of rules. Moreover even if the contexts of such operators are all referentially opaque, in Quine's sense [1] of being exempt from the substitutivity of truth-functional biconditionals, some are, as it were, opaquer than others, in being exempt from the substitutivity of certain non-truth-functional biconditionals as well.

The global function on finite systems is a new concept defined by Gurevich in [1] and discussed in [2] and [3]. In the last ten years this concept has become more and more useful in computer science and logic. Gurevich also pointed out the importance of global functionals on finite systems. In this paper we will give a brief introduction to the concepts of global functions and global functionals on finite systems.

In studying the natural number system N = 〈N, +,0〉 we often refer to its functions and functionals. There are a lot of books and papers in this area. Kleene in [4] gave a detailed introduction to the recursive functions of N. The functionals of N are normally very difficult to compute because here we need to tell the machine what the input function is, which is not very easy to do. In developing the theory of finite systems the functions and functionals are also very useful. For computing the functionals in finite systems we can take the entire graph of a function as the input, which is not possible in N. We will discuss recursive functions and functionals for finite systems. The definitions of recursive functions are very similar to the case in N, but we will have a very different situation. In N the number of elements is infinite. The number of all possible functions from N to N is the continuum. In a finite system the number of all possible functions is finite. It seems that there is no necessity to define the global functions.

With reservations, one can think of abstract algebra as the study of what consequences can be drawn from the axioms associated with certain concrete algebraic structures. Two important examples of such concrete algebraic structures are the integers and the rational numbers. The integers and the rational numbers have two properties which are not in general mirrored in the abstract axiom systems associated with them. That is, the integers and the rational numbers both have effectively computable metrics and their algebraic operations are effectively computable. The study of abstract algebraic systems which possess effectively computable algebraic operations has produced many interesting results. One can think of a computable algebraic structure as one whose elements have been labeled by the set of positive integers and whose operations are effectively computable. The formal definition of computable local integral domain will be given in §3. Some specific computable structures which have been studied are the integers, the rational numbers, and the rational numbers with p-adic valuation. Computable structures were studied in general by Rabin [12]. This paper concerns computable local integral domains as exemplified by the local integral domain Zp. Zp is the localization of the integers with respect to the maximal prime ideal generated by the positive prime p. We should note that the concept of local integral domain is not first order.

Let the ordered pair (Q, M) stand for a local ring, where Q is the local ring and M is the unique maximal prime ideal of Q. Since most of my results are proving the existence of certain effective procedures, the assumption that Q has a principal maximal ideal M (rather than M has n generators) greatly simplifies many of the proofs.

Plusieurs auteurs ont déjà considéré l'ultraproduit et l'ultrapuissance en prenant certaines, et non forcément toutes les fonctions qui, à chaque indice i, associent un élément de la base de la relation de rang i (voir par exemple [6]). La généralisation ici considérée, appelée ultraproduit complet (voir §3) est définie à l'aide des isomorphismes locaux. On y retrouve l'essentiel de la théorie de l'ultraproduit classique avec évidemment des exemples plus variés.

Each Girard quantale (i.e., commutative quantale with a selected dualizing element) provides a support for a semantics for linear propositional formulas (but not for linear derivations). Several constructions of Girard quantales are known. We give two more constructions, one using an arbitrary partially ordered monoid and one using a partially ordered group (both commutative). In both cases the semantics can be controlled be a relation between pairs of elements of the support and formulas. This gives us a neat way of handling duality.