Lattices are regular arrangements of points in n-dimensional space, whose study appeared in the 19th century in both number theory and crystallography. Since the appearance of the celebrated Lenstra-Lenstra-Lovv asz lattice basis reduction algorithm twenty years ago, lattices have had surprising applications in cryptology. Until recently, the applications of lattices to cryptology were only neg...

Open lattices are characterized by low-volume-fraction arrangements of building blocks, low coordination number, and open spaces between building blocks. The self-assembly of these lattices faces the challenge of mechanical instability due to their open structures. We theoretically investigate the stabilizing effects of entropy in the self-assembly of open lattices from patchy particles. A prel...

Recently, several authors extended Priestley duality for distributive lattices [9] to other classes of algebras, such as, e.g. distributive lattices with operators [7], MV -algebras [8], MTL and IMTL algebras [2]. In [5] necessary and sufficient conditions for a normally presented variety to be naturally dualizable, in the sense of [6], i.e. with respect to a discrete topology, have been provid...

This formalization introduces and collects some algebraic structures based on lattices and complete lattices for use in other developments. The structures introduced are modular, and lattice ordered groups. In addition to the results proved for the new lattices, this formalization also introduces theorems about latices and complete lattices in general.

The goal of this survey is to study some classes of groups determined by different types of complementation of their normal subgroup lattices. The groups whose normal subgroup lattices are Stone lattices, boolean lattices or ortholattices will be also investigated. MSC (2000): Primary 20D30, Secondary 20E15, 06C15, 06D15.

Wild, M., Cover preserving embedding of modular lattices into partition lattices, Discrete Mathematics 112 (1993) 207-244. When is a finite modular lattice couer preserving embeddable into a partition lattice? We give some necessary, and slightly sharper sufficient conditions. For example, the class of cover preserving embeddable modular lattices strictly contains the class of acyclic modular l...

Categorical skew lattices are a variety of skew lattices on which the natural partial order is especially well behaved. While most skew lattices of interest are categorical, not all are. They are characterized by a countable family of forbidden subalgebras. We also consider the subclass of strictly categorical skew lattices.

The asymptotic lattices and their transformations are studied within the line geometry approach. It is shown that the discrete asymptotic nets are represented by isotropic congruences in the Plücker quadric. On the basis of the Lelieuvre-type representation of asymptotic lattices and of the discrete analog of the Moutard transformation, it is constructed the discrete analog of the W–congruences...

We propose a general framework to study constructions of Euclidean lattices from linear codes over finite fields. In particular, we prove general conditions for an ensemble constructed using linear codes to contain dense lattices (i.e., with packing density comparable to the Minkowski-Hlawka lower bound). Specializing to number field lattices, we obtain a number of interesting corollaries for i...