The answer is yes, if ... This note attempts to give amplification to the above statement, while at the same time arriving at a reasonable description of this lattice. The main theorem of the paper is no doubt the assertion that the lattice of torsion classes of lattice-ordered groups is completely distributive. The proof of this theorem depends on the notion of a value selector, and should not...

A positively convex module is a non-empty set closed under positively convex combinations but not necessarily a subset of a linear space. Positively convex modules are a natural generalization of positively convex subsets of linear spaces. Any positively convex module has a canonical semimetric and there is a universal positively affine mapping into a regularly ordered normed linear space and a...

Given a quasi-concave-convex function f : X × Y → R defined on the product of two convex sets we would like to know if infY supX f = supX infY f . In [4] we showed that that question is very closely linked to the following “reconstruction” problem: given a polytope (i.e. the convex hull of a finite set of points) X and a family F of subpolytopes of X, we would like to know if X ∈ F, knowing tha...

Let K and L be compact convex sets in Rn. The following two statements are shown to be equivalent: (i) For every polytope Q ⊆ K having at most n+ 1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 ≤ d ≤ n − 1. It is also shown that the following two statements are equivalent: (i) For every polytope Q ⊆ K having at most d+ 1 vertices, L contains a translate of Q. (i...

O-minimal expansions of real closed fields Dissertation Zur Erlangung des Doktorgrades der Naturwissenschaften aus Regensburg 1996 Introduction We are concerned with o-minimal theories. The main result (the Box Theorem 17.3) is about polynomially bounded, o-minimal expansions T of real closed fields with archimedean prime model. In this Introduction, T always denotes such a theory-the reader ma...

We show that, if a building is endowed with its complete system of apartments, and if each panel is contained in at least four chambers, then the intersection of two apartments can be any convex subcomplex contained in an apartment. This combinatorial result is particularly interesting for lower dimensional convex subcomplexes of apartments, where we definitely need the assumption on the four c...

In this paper we discuss symmetrically self-dual spaces, which are simply real vector spaces with a symmetric bilinear form. Certain subsets of the space will be called q-positive, where q is the quadratic form induced by the original bilinear form. The notion of q-positivity generalizes the classical notion of the monotonicity of a subset of a product of a Banach space and its dual. Maximal q-...

1. Introduction. Let p be a prime number such that q — l{p — l) is also a prime. Let Q, be the set of symbols 1, • • • , p, and ® be a non-solvable transitive permutation group on 12. Such permutation groups were first considered by Galois in 1832 [I, §327; III, §262]: if the linear fractional group LF 2 (l) over the field of I elements, where I is a prime number not smaller than five, contains...

Both spaces were introduced and shown to be one-dimensional but totally disconnected by Paul Erdős [9] in 1940. This result together with the obvious fact that E and Ec are homeomorphic to their squares make these spaces important examples in Dimension Theory. Both E and Ec are universal spaces for the class of almost zero-dimensional spaces; see [7, Theorem 4.15]. A subset of a space is called...