![]() ![]() ![]() The term Verband refers to an algebra (in the sense of universal algbera) with a join and a meet operation. ![]() In general I would see the two main classes, that can are distinguished in German language. So does it means that lattice represent a path, where some specific condition held or? Mostly I am interested application of lattice in complex analysis Skew lattice, a noncommutative generalization of a lattice Lattice model (finance), a method for evaluating stock options that divides time into discrete intervals Lattice multiplication, a multiplication algorithm suitable for hand calculation Lattice graph, a graph that can be drawn within a repeating arrangement of pointsīethe lattice, a regular infinite tree structure Lattice (module), a module over a ring embedded in a vector space over a field Lattice (discrete subgroup), a discrete subgroup of a topological group with finite covolume Lattice (group), a repeating arrangement of points Lattice (order), a partially ordered set with unique least upper bounds and greatest lower bounds Then general definition of lattice from group theory and from different branches of mathematics Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions. This can be justified by the intention to make the report as complete as possible.I would like to understand meaning of lattice in mathematics, for example let us consider its application, first one is Elliptic function: Finally, the basic notions (for group theory see , , for lattice theory ) and a brief survey of results in this field are contained in the new edition of Kurosh's monograph Under the circumstances there is naturally some overlap between this work and the above articles. General surveys of the results (up to 1961) for infinite soluble, nilpotent and radical groups are contained in the papers of Kontorovich, Pekelis, and Starostin and Plotkin . The structures of various classes of finite groups, their lattice isomorphisms and homomorphisms, and many related questions are treated in Suzuki's comprehensive monograph . The main attention is given to infinite groups. The present survey (without claiming to exhaust the whole material) is devoted to the so-called lattice-theoretical questions of group theory and contains recent results. These problems seem to be difficult, and apparently new ideas and mathematical techniques are needed for their solution. There are many interesting results in this field, but also many important unsolved problems. The investigation of connections between the structure of a group and the structure of its lattice of subgroups has become one of the main general approaches in the study of groups. This explains to what extent purely set-theoretical operations given on the set of all subgroups of a group $G$ (this set forms a lattice $S(G)$) determine the properties of the group operation itself. However, in a number of investigations essentially the converse problems are solved: to study the influence of the properties of the set of all subgroups of a group on the properties of its elements. Usually, properties of a group are obtained from the properties of its elements. The study of group properties and group structure can proceed from various initial principles. ![]()
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