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 nonlogical 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 ørst-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 inønitary 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.

Abstract. We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF C ∗-algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups. 1.

this paper is to demonstrate how sharp that knife-edge is, and thus to establish some limits on what such qualitative spatial description languages might be like. Specifically, we show that, once we can represent the property of convexity and the part-whole relation---modest assumptions by any standards--- n-tuples of real polygons are completely determined by the sets of formulas they satisfy upto the fixing of three reference points. To be sure, we are not the first to express skepticism about the possibility of such languages, but this is, as far as we are aware, the first time such skepticism has been put on so firm a mathematical footing.

For infinite societies, Fishburn (1970), Kirman and Sondermann (1972), and Armstrong (1980) gave a nonconstructive proof of the existence of a social welfare function satisfying Arrow’s conditions (Unanimity, Independence, and Nondictatorship). This paper improves on their results by (i) giving a concrete example of such a function, and (ii) showing how to compute, from a description of a profile on a pair of alternatives, which alternative is socially preferred under the function. The introduction of a certain “oracle ” resolves Mihara’s impossibility result (1997) about computability of social welfare functions.

Let (P, <=) be a partially ordered set. We define the poset Boolean algebra of P, and denote it by F (P). The set of generators of F(P) is {x p : p P}, and the set of relations is q}. We say that a Boolean algebra B is well-generated, if B has a sublattice G such that G generates B and G is well-founded. A well-generated algebra is superatomic...

Abstract. For a given class T of compact Hausdorff spaces, let Y(T) denote the class of ℓ-groups G such that for each g ∈ G, the Yosida space Y(g)of g belongs to T.Conversely,ifR is a class of ℓ-groups, then T(R) stands for the class of all spaces which are homeomorphic to a Y(g) for some g ∈ G ∈ R. The correspondences T ↦ → Y(T) and R ↦ → T(R) are examined with regard to several closure properties of classes. Several sections are devoted to radical classes of ℓ-groups whose Yosida spaces are zero-dimensional. There is a thorough discussion of hyper-projectable ℓ-groups, followed by presentations on Y(e.d.),wheree.d. denotes the class of compact extremally disconnected spaces, and, for each regular uncountable cardinal κ, the class Y(discκ), where discκ stands for the class of all compact κ-disconnected spaces. Sample results follow. Every strongly projectable ℓ-group lies in Y(e.d.). Theℓ-group G lies in Y(e.d.) if and only if for each g ∈ GY(g)is zero-dimensional and the Boolean algebra of components of g, comp(g), is complete. Corresponding results hold for Y(discκ). Finally, there is a discussion of Y(F), with F standing for the class of compact F-spaces. It is shown that an Archimedean ℓ-group G is in Y(F) if and only if, for each pair of disjoint countably generated polars P and Q, G = P ⊥ + Q ⊥.

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Abstract. This paper continues the investigation into Krull-style dimensions in algebraic frames. Let L be an algebraic frame. dim(L) is the supremum of the lengths k of sequences p0 < p1 < · · · < pk of (proper) prime elements of L. Recently, Th. Coquand, H. Lombardi and M.-F. Roy have formulated a characterization which describes the dimension of L in terms of the dimensions of certain boundary quotients of L. This paper gives a purely frame-theoretic proof of this result, at once generalizing it to frames which are not necessarily compact. This result applies to the frame Cz(X) of all z-ideals of C(X), provided the underlying Tychonoff space X is Lindelöf. If the space X is compact, then it is shown that the dimension of Cz(X) is at most n if and only if X is scattered of Cantor-Bendixson index at most n + 1. If X is the topological union of spaces Xi, then the dimension of Cz(X) is the supremum of the dimensions of the Cz(Xi). This and other results apply to the frame of all d-ideals Cd(X) of C(X), however, not the characterization in terms

Abstract. ATR0 is the natural subsystem of second-order arithmetic in which one can develop a decent theory of ordinals. We investigate classes of structures which are in a sense the “well-founded part ” of a larger, simpler class, for example, superatomic Boolean algebras (within the class of all Boolean algebras). The other classes we study are: well-founded trees, reduced Abelian p-groups, and countable, compact topological spaces. Using computable reductions between these classes, we show that Arithmetic Transfinite Recursion is the natural system for working with them: natural statements (such as comparability of structures in the class) are equivalent to ATR0. The reductions themselves are also objects of interest. 1.

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