understanding lattice in detailed

Question

I would like to understand meaning of lattice in mathematics, for example let us consider its application, first one is Elliptic function:

In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions. 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

Then general definition of lattice from group theory and from different branches of mathematics

Lattice (order), a partially ordered set with unique least upper bounds and greatest lower bounds
Lattice (group), a repeating arrangement of points
Lattice (discrete subgroup), a discrete subgroup of a topological group with finite covolume
Lattice (module), a module over a ring embedded in a vector space over a field

Lattice graph, a graph that can be drawn within a repeating arrangement of points

Bethe lattice, a regular infinite tree structure

Lattice multiplication, a multiplication algorithm suitable for hand calculation

Lattice model (finance), a method for evaluating stock options that divides time into discrete intervals

Skew lattice, a noncommutative generalization of a lattice

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

Answer 1

In general I would see the two main classes, that can are distinguished in German language.

• The term Verband refers to an algebra (in the sense of universal algbera) with a join and a meet operation. There is a duality between lattice ordered sets (in German: “verbandsgeordnete Mengen”) and lattices.
• The term Gitter refers to a periodic structure, usually periodically arranged lines in one or more direction or the points of their intersection. This kind of lattice is typically linked to the embedding of a module in a vector space. This would also fit into your group category. E.g. every torsion free abelian group is a $\mathbb Z$ module. This kind of lattice is also behind the lattice multiplication where the notion “lattice” is not taken from mathematical but from natural language. This should be true also for the financial lattice model, which considers the grid points instead of integrating over an interval.

In case of complex analysis you are probably interested in the second one if you are in the approximation of certain point sets by similar sets with sufficiently regular borders. On the other hand lattice theory in the first sense can be used as a tool for proofs. In that case it serves more or less as a data structure in which you can put certain objects and prove that the desired object is also in that structure or not.