Application of the theory of algebraic systems for creating a hierarchy of solid structures formed under equilibrium and nonequilibrium conditions

A unified hierarchy is proposed for molecular and solid structures formed under equilibrium (ideal crystals) or nonequilibrium conditions (real crystals, fractally ordered crystalline, quasicrystalline, and amorphous solids, as well as composite solid materials that are aperiodic on an atomic-molecular level but are periodic on a macroscopic level). The construction of this hierarchy is based on applying the theory of algebraic systems (groups, rings, and fields) to the multiplication of an initial structure in space depending on an inflation coefficient (numbers) expressed in the general form \(Q = (n + m\sqrt l )/k\). Examples are presented of molecular and polymer structures described by groups or rings, fractally ordered solids whose structures are described by fields, and solids with damped or self oscillations in their composition, whose structures are described by fields or periodic rings of fields with complex spatial multiplication factors.