# understanding lattice in detailed

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

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Lattice is just one of the overloaded terms in math. Other common ones (in no particular order) are: normal, degree, order, basis... We seek to do with words from plain English rather than inventing new ones - collisions are inevitable. –  Jyrki Lahtonen Sep 7 '13 at 12:53
could you clarify what did you mean? –  dato datuashvili Sep 7 '13 at 13:04

• 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.