We prove some restrictions on the possible cofinalities of ultrapowers of the natural numbers with respect to ultrafilters on the natural numbers. The restrictions involve three cardinal characteristics of the continuum, the splitting number s, the unsplitting number r, and the groupwise density number g. We also prove some related results for reduced powers with respect to filters other than ultrafilters.

We compare several closely related continuum invariants, i.e., a, ae. ap in two forcing models. And we shall ask some open questions in this field.

In this paper we show that relativized versions of relation set algebras and cylindric set algebras have undecidable equational theories if we include coordinatewise versions of the counting operations into the similarity type. We apply these results to the guarded fragment of first-order logic.

We prove that in a nilpotent-by-finite stable group an equation that holds generically holds everywhere. Combining this result with results of Wagner and Bryant, we conclude that a soluble-by-finite stable group of generic exponent n has exponent n.

If α and β are ordinals, α ≤ β, and β ≰ α then α+ 1 < β. The first result of this paper shows that the restriction of this statement to countable well orderings is provably equivalent to ACA0, a subsystem of second order arithmetic introduced by Friedman. The proof of the equivalence is reminiscent of Dekker's construction of a hypersimple set. An application of the theorem yields the equivalence of the set comprehension scheme ACA0 and an arithmetical transfinite induction scheme.

Let M be a countably infinite ω-categorical structure. Consider Aut(M) as a complete metric space by defining d(g, h) = Ω{2−n: g (xn) ≠ h(xn) or g−1 (xn) ≠ h−1 (xn)} where {xn : n ∈ ω} is an enumeration of M An automorphism α ∈ Aut(M) is generic if its conjugacy class is comeagre. J. Truss has shown in [11] that if the set P of all finite partial isomorphisms contains a co-final subset P1 closed under conjugacy and having the amalgamation property and the joint embedding property then there is a generic automorphism. In the present paper we give a weaker condition of this kind which is equivalent to the existence of generic automorphisms. Really we give more: a characterization of the existence of generic expansions (defined in an appropriate way) of an ω-categorical structure. We also show that Truss' condition guarantees the existence of a countable structure consisting of automorphisms of M which can be considered as an atomic model of some theory naturally associated to M. We do it in a general context of weak models for second-order quantifiers.

The author thanks Ludomir Newelski for pointing out a mistake in the first version of Theorem 1.2 and for interesting discussions. Also, the author is grateful to the referee for very helpful remarks.

Recently Lafont [6] showed the finite model property for the multiplicative additive fragment of linear logic (MALL) and for affine logic (LLW), i.e., linear logic with weakening. In this paper, we shall prove the finite model property for intuitionistic versions of those, i.e. intuitionistic MALL (which we call IMALL), and intuitionistic LLW (which we call ILLW). In addition, we shall show the finite model property for contractive linear logic (LLC), i.e., linear logic with contraction. and for its intuitionistic version (ILLC). The finite model property for related substructural logics also follow by our method. In particular, we shall show that the property holds for all of FL and GL−-systems except FLc and of Ono [11], that will settle the open problems stated in Ono [12].

A notion of complexity for an arbitrary structure was defined in the book of Poizat Les petits cailloux (1995): we can define P and NP problems over a differential field K. Using the Witness Theorem of Blum et al., we prove the P-stability of the theory of differential fields: a P problem over a differential field K is still P when restricts to a sub-differential field k of K. As a consequence, if P = NP over some differentially closed field K, then P = NP over any differentially closed field and over any algebraically closed field.

In this paper a class of simple theories, called the low theories is developed, and the following is proved.

Theorem. Let T be a low theory, A a set and a, b elements realizing the same strong type over A. Then, a and b realize the same Lasear strong type over A.

The class of all Artinian local rings of length at most l is ∀2-elementary, axiomatised by a finite set of axioms τtl. 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 oτl of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory τtl is companionable, with model-companion oτl.

In Barwise and van Benthem [6], the authors give a general method for obtaining interpolation and preservation theorems for fragments of L∞ω, those for which there is a co-inductive pebble game Γ characterizing equivalence in the logic. The method is exemplified by an analysis of the following fragments: L∞ω itself, its existential fragment , its positive fragment , the k-variable fragment (and its existential and positive subfragments) and the modal fragment (and its existential and positive subfragments).

While most of their method is general, there is one part (showing that Γ has the Scott property relative to the fragment) that required a case-by-case analysis. The purpose of our paper is to replace this case-by-case analysis by a general theorem, and to illustrate this method by obtaining their kinds of results for some additional fragments of L∞ω.

Our general problem can be stated in the following way: Given a “nice” fragment F of L∞ω (one satisfying some natural closure conditions), find a pebble game characterization Γ of “preservation of F-formulas” and prove that Γ has the Scott property with respect to F. Applying the Abstract Interpolation Theorem from [6], we can conclude that F has Γ-interpolation, and the corresponding preservation result. In this paper, we shall give an answer to this question. (Our answer is “sufficient” but we don't know if our conditions are necessary.)

We show that the variety of Heyting algebras has finitary unification type. We also show that the subvariety obtained by adding it De Morgan law is the biggest variety of Heyting algebras having unitary unification type. Proofs make essential use of suitable characterizations (both from the semantic and the syntactic side) of finitely presented projective algebras.

Preservation and interpolation results are obtained for L∞ω and sublogics ⊆ L∞ω such that equivalence in can be characterized by suitable back-and-forth conditions on sets of partial isomorphisms.