Abstract algebra
lastmod 2014-10-29created 2014-10-291 edits
Abstract algebra (抽象代數學) is a branch of mathematics. Abstract algebra studies groups, rings, fields, modules, vector spaces, and algebras. The name "abstract algebra" was coined in the 20th century. Before then, "abstract algebra" and "elementary algebra", which deals with the four arithmetic operations, solving equations, and computing with real and complex numbers, were lumped together and called algebra. So the term abstract algebra was newly coined to make it clear that elementary algebra, which people who did not major in mathematics generally think of when they think of algebra, and the area covered by abstract algebra are different. In recent works, however, the distinction is becoming blurred again.
Historically, algebraic structures first arose in several other areas of mathematics, and only after they were specified axiomatically did they find their place and begin to be studied in abstract algebra. Because of this, abstract algebra has developed numerous connections with all other branches of mathematics.
Modern mathematics and mathematical physics borrow the concepts of abstract algebra especially heavily. For example, theoretical physics makes deep use of Lie algebras, a branch of abstract algebra. Algebra is also applied to many other branches of mathematics through algebraic number theory, algebraic topology, and algebraic geometry. Representation theory is a discipline that studies the algebraic structures covered by abstract algebra in greater depth. Examples of algebraic structures with a single binary operation:
- Semigroup
- Monoid: a semigroup with an identity element
- Quasigroup
- Group
More complex examples:
- Rings and fields
- Modules and vector spaces
- Associative algebras and Lie algebras
- Lattices and Boolean algebras
The properties common to all these algebraic structures are studied in category theory. Category theory provides a formal means of comparing different algebraic structures and studying the correspondences between them.