A dilator D is a functor from ON to itself commuting with direct limits and pull-backs. A dilator D is a flower iff D(x) is continuous. A flower F is regular iff F(x) is strictly increasing and F(f)(F(z)) = F(f(z)) (for f ϵ ON(x,y), z ϵ X).

Equalization is the following axiom: if F, G ϵ Flr (class of regular flowers), then there is an H ϵ Flr such that F ° H = G ° H. From this we can deduce that if ℱ is a set ⊆ Flr, then there is an H ϵ Flr which is the smallest equalizer of ℱ (it can be said that H equalizes ℱ iff for every F, G ϵ ℱ we have F ° H = G ° H). Equalization is not provable in set theory because equalization for denumerable flowers is equivalent to -determinacy (see a forthcoming paper by Girard and Kechris).

Therefore it is interesting to effectively find, by elementary means, equalizers even in the simplest cases. The aim of this paper is to prove Girard and Kechris's conjecture: “ is the (smallest) equalizer for Flr < ω” (where Flr < ω denotes the set of finite regular flowers). We will verify that is an equalizer of Flr < ω; we will sketch the proof that it is the smallest one at the end of the paper. We will denote by H.

On sait que par le choix d’ un langage suffisamment complexe, toute structure peut admettre une élimination des quantificateurs, malheureusement cette extension du langage peut nous éloigner des phénomènes algébriques. Nous allons nous intéresser à l’ élimination des quantificateurs pour des paires de corps. Dans le cas des paires de corps algébriquement clos et des paires denses de corps réel clos, on obtient une élimination en ajoutant au langage des prédicats ayant une signification algébrique: on peut les exprimer en disant que pour deux ensembles algébriques et donnés, il existe des points du sous-corps rationnels pour et pas pour , ou qu’ un ensemble semi-algébrique donné a des points rationnels sur le sous-corps. Robinson avait déjà abordé de façon informelle le cas des paires denses de corps réel-clos (cf. [Ro 2, p. 198]). Partant du langage des paires de corps ordonnés, enrichi de symboles de relations correspondant à l’ indépendance algébrique, il proposait d’ ajouter pas à pas des fonctions de Skolem Herbrand pour faire disparaitre les quantificateurs existentiels des formules, mais sans préciser le langage obtenu. Ici nous approchons le problème différemment en explicitant dès le départ le langage utilisé.

Grâce à ces résultats nous pourrons étudier le cas des paires séparées de corps réels clos ainsi que des paires de corps valués henseliens. En élargissant la définition d’ ensemble algébrique à tous les symboles du langage, les prédicats relationnels ajoutés ont la même signification que dans le cas des paires de corps algébriquement clos.

En comparant les techniques employées ici avec celles déjà utilisées dans [K], [B], [D 1] et [L], on remarque qu’il est possible de traîter une grande partie de l’ étude (complétude, modèle complétude, élimination des quantificateurs) des paires de corps algébriquement clos, réel-clos ou henseliens en se basant sur des prolongements d’ isomorphismes entre sous-structures dénombrables où l’ une des deux est contenue dans une structure ω1-saturée.

We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.

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.

A Boolean product construction is used to give examples of existentially closed algebras in the universal Horn class ISP(K) generated by a universal class K of finitely subdirectly irreducible algebras such that Γa(K) has the Fraser-Horn property. If ⟦a ≠ b⟧ ∩ ⟦c ≠ d ⟧ = ∅ is definable in K and K has a model companion of K-simple algebras, then it is shown that ISP(K) has a model companion. Conversely, a sufficient condition is given for ISP(K) to have no model companion.

We prove that the only implications between four notions for Σ½-sets of reals are

Let F be the class of complete, finitely axiomatizable ω-categorical theories. It is not known whether there are simple theories in F. We prove three results of the form: if T ∈ F has a sufficently well-behaved definable set J, then T is not simple. (In one case, we actually prove that T has the strict order property.) All of our arguments assume that the definable set J satisfies the Mazoyer hypothesis, which controls how an element of J can be algebraic over a subset of the model. For every known example in F, there is a definable set satisfying the Mazoyer hypothesis.

The problem of coding relations by means of a single binary relation is well known in the mathematical literature. It was considered in interpretation theory, and also in connection with investigations of decidability of elementary theories. Using various constructions (see, e.g., [2,6], proofs of Theorem 11 in [7] and Theorem 16.51 in [3]), for any model for a countable language, one can construct a model for ℒp (a language with a single binary relation symbol ) in which is interpretable. Each of the mentioned constructions has the same weak point: the universe of is different than the universe of . In [4] we have shown that, in the infinite case, one can eliminate this defect, i.e., for any infinite , we have constructed a model , having the same universe as , in which is elementarily definable. In all constructions mentioned above, it appears that formulas, which define in ( in ), are very complicated. In the present paper, another construction of a model for ℒp is given.