We introduce the concept of a locally finite abstract elementary class and develop the theory of excellence (with respect to disjoint ( , k)-amalgamation) for such classes. From this we find a family of complete L!1,! sentences r that a) homogeneously characterizes @r (improving results of Hjorth [12] and Laskowski-Shelah [14] and answering a question of [22]), while b) the r provide the first...

The goal of this paper is the construction of measures μ on R(n)enjoying three conflicting but fortunately compatible properties: (i) μ is a sum of weighted Dirac masses on a locally finite set, (ii) the Fourier transform μ f μ is also a sum of weighted Dirac masses on a locally finite set, and (iii) μ is not a generalized Dirac comb. We give surprisingly simple examples of such measures. These...

The lattice of closed subsets of a set under such a closure operator is semimodular. Perhaps the best known example of a closure operator satisfying the exchange principle is the closure operator on a vector space W where for X ___ W we let C(X) equal the span of X. The lattice of C-closed subsets of W is isomorphic to Con(W) in a natural way; indeed, if Y _~ W x W and Cg(Y) denotes the congrue...