Let k ⊂ K be a field extension, where K is an algebraically closed field of any characteristic and k is the prime field. Recall the following property of Hilbert Schemes (see, for example, [1], Proposition 1.16): Suppose ⊂ × S is a flat family of closed subschemes of parametrised by a scheme S/k. Then for every closed subscheme Z ⊂ in , if [Z] denotes the Hilbert point of Z in Hilb() then the residue field of Hilb() at [Z] is the minimal field of definition for Z. Intuitively, this says that as a family parametrised by Hilb(), each fibre of lies above a point whose “co-ordinates” generate its minimal field of definition.

In the following note we are concerned with a more naive form of the above situation, for which we wish to give an elementary account. Suppose ϕ(x,y) is a system of polynomial equations over k (in variables x = (x1,…, xm) and parameters y = (y1, …, yn)), such that

is a family of (possibly reducible) affine varieties in Km. Each nonempty member of this family has a unique minimal field of definition. The question arises as to whether it is possible to express this family of varieties using parameters that come, pointwise, from these minimal fields of definition. That is, is there a system of polynomial equations ψ(x, z) over k, such that each ψ(x, b) with b ∈ KN is of the form Va for some a ∈ Kn; and such that each Va ∈ is defined by ψ(x, b) for some b ∈ KN whose coordinates generate the minimal field of definition for Va? Moreover, we would like b to be obtained definably from a.

In recent years, there has been renewed interest in the development of formal languages for describing mereological (part-whole) and topological relationships between objects in space. Typically, the non-logical primitives of these languages are properties and relations such as ‘x is connected’ or ‘x is a part of y’, and the entities over which their variables range are, accordingly, not points, but regions: spatial entities other than regions are admitted, if at all, only as logical constructs of regions. This paper considers two first-order mereotopological languages, and investigates their expressive power. It turns out that these languages, notwithstanding the simplicity of their primitives, are surprisingly expressive. In particular, it is shown that infinitary versions of these languages are adequate to express (in a sense made precise below) all topological relations over the domain of polygons in the closed plane.

We give a new proof of the classification of ℵ0-categorical quasivarieties by using Morita equivalence to reduce to term minimal quasivarieties.

Here we investigate the classes of representable directed cylindric algebras of dimension α introduced by Németi [12]. can be seen in two different ways: first, as an algebraic counterpart of higher order logics and second, as a cylindric algebraic analogue of Quasi-Projective Relation Algebras. We will give a new, “purely cylindric algebraic” proof for the following theorems of Németi: (i) is a finitely axiomatizable variety whenever α ≥ 3 is finite and (ii) one can obtain a strong representation theorem for if one chooses an appropriate (non-well-founded) set theory as foundation of mathematics. These results provide a purely cylindric algebraic solution for the Finitization Problem (in the sense of [11]) in some non-well-founded set theories.

In this paper we set out the basic model theory of differential fields of characteristic 0, which have finitely many commuting derivations. We give axioms for the theory of differentially closed differential fields with m derivations and show that this theory is ω-stable, model complete, and quantifier-eliminable, and that it admits elimination of imaginaries. We give a characterization of forking and compute the rank of this theory to be ωm + 1.

It is possible to define a combinatorial closure on alternating bilinear maps with few relations similar to that in [2]. For the ℵ0-categorical case we show that this closure is part of the algebraic closure.

It is shown that the many-dimensional modal logic Kn, determined by products of n-many Kripke frames, is not finitely axiomatisable in the n-modal language, for any n > 2. On the other hand, Kn is determined by a class of frames satisfying a single first-order sentence.

We consider the monadic second-order theory of linear order. For the sake of brevity, linearly ordered sets will be called chains.

Let = ⟨A <⟩ be a chain. A formula ø(t) with one free individual variable t defines a point-set on A which contains the points of A that satisfy ø(t). As usually we identify a subset of A with its characteristic predicate and we will say that such a formula defines a predicate on A.

A formula (X) one free monadic predicate variable defines the set of predicates (or family of point-sets) on A that satisfy (X). This family is said to be definable by (X) in A. Suppose that is a subchain of = ⟨B, <⟩. With a formula (X, A) we associate the following family of point-sets (or set of predicates) {P : P ⊆ A and (P, A) holds in } on A. This family is said to be definable by in with at the background.

Note that in such a definition bound individual (respectively predicate) variables of range over B (respectively over subsets of B). Hence, it is reasonable to expect that the presence of a background chain allows one to define point sets (or families of point-sets) on A which are not definable inside .

We make a more systematic study of the van Douwen diagram for cardinal coefficients related to combinatorial properties of partitions of natural numbers.