For finite semidistributive lattices the map κ gives a bijection between sets of completely join-irreducible elements and meet-irreducible elements. Here we study -map in context torsion classes. It is well-known that lattice classes for an artin algebra semidistributive, but general it far from finite. We show well-defined on set elements, even when infinite. then extend to which have canonica...

We study combinatorics of billiard partitions which arose recently in the description periodic trajectories ellipsoidal billiards d-dimensional Euclidean and pseudo-Euclidean spaces. Such uniquely codify sets caustics, up to their types, generate trajectories. The period a trajectory is largest part while winding numbers are remaining summands corresponding partition. In order take into account...

Additive combinatorics is the branch of combinatorics where the objects of study are subsets of the integers or of other abelian groups, and one is interested in properties and patterns that can be expressed in terms of linear equations. More generally, arithmetic combinatorics deals with properties and patterns that can be expressed via additions and multiplications. In the past ten years, add...