Introduction
============

.. _aggr-intro:

This part of the Handbook comprises seven chapters on aggregate objects in Magma
as well as a chapter on maps.

Sets, sequences, tuples, and lists are the four main types of aggregates, and 
each has its own specific purpose. *Sets* are used to collect objects that are 
elements of some common structure, and the most important operation is to test
element membership. *Sequences* also contain objects of a common structure, but 
here the emphasis is on the ordering of the objects, and the most important 
operation is that of accessing (or modifying) elements at given positions. Sets 
will contain at most one copy of any element, whereas sequences may contain 
arbitrarily many copies of the same object.

*Enumerated* sets and sequences are of arbitrary but finite length and will 
store all elements explicitly (with the exception of arithmetic progressions),
while *formal* sets and sequences may be infinite, and use a Boolean function
to test element membership. *Indexed* sets are a hybrid form of sets allowing 
indexing like sequences.

Elements of *Cartesian products* of structures in Magma will be called *tuples*; 
they are of fixed length, and each coefficient must be in the corresponding 
structure of the defining Cartesian product. *Lists* are arbitrary finite 
ordered collections of objects of any type, and are mainly provided to the user 
to store assorted data to which access is not critical.