Assuming the Gowers Inverse conjecture and the Möbius conjecture for the finite parameter s, Green-Tao verified Dickson’s conjecture for lattices which are ranges of linear maps of complexity at most s. In this paper, we reformulate GreenTao’s theorem on Dickson’s conjecture, and prove that, if L is the range of a linear map of complexity s, and L1 is a sublattice of L of finite index, then L1 ...

In this paper, a characterization of the language varieties and congruence varieties corresponding toM -solid pseudovarieties is presented. Taking into account the isomorphisms of the Eilenberg-type correspondences, each complete sublattice of pseudovarieties corresponds to a complete sublattice of language varieties, as well as another one of congruence varieties. For the varieties of tree lan...

We study the statistical physical properties of (discretized) “random surfaces,” which are random functions from Z (or large subsets of Z) to E, where E is Z or R. Their laws are determined by convex, nearest-neighbor, gradient Gibbs potentials that are invariant under translation by a full-rank sublattice L of Z; they include many discrete and continuous height function models (e.g., domino ti...

The aim of this paper is to introduce a new notion of L-fuzzy compactness in L-fuzzy topological spaces, which is a generalization of Lowen’s fuzzy compactness in L-topological spaces. The union of two L-fuzzy compact L-sets is L-fuzzy compact. The intersection of an L-fuzzy compact L-set G and an L-set H with T(H) = ⊤ is L-fuzzy compact. The L-fuzzy continuous image of an L-fuzzy compact L-set...

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...