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Algebraic structures appear in most branches of mathematics, and students can encounter them in many different ways. Beginning study: In American universities, groups, vector spaces and fields are generally the first structures encountered in subjects such as linear algebra. They are usually introduced as sets with certain axioms. Advanced study: Abstract algebra studies properties of specific algebraic structures. Universal algebra studies algebraic structures abstractly, rather than specific types of structures. Varieties Category theory studies interrelationships between different structures, algebraic and non-algebraic. To study a non-algebraic object, it is often useful to use category theory to relate the object to an algebraic structure. Example: The fundamental group of a topological space gives information about the topological space. |
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