User:Phlsph7/Algebra - Definition

Definition
Algebra is the branch of mathematics that studies algebraic operations and algebraic structures. An algebraic structure is a non-empty set of mathematical objects, such as the real numbers, together algebraic operations defined on that set, such addition and multiplication. Algebra explores the laws and general characteristics of algebraic structures and what types of algebraic structures there are. Within certain algebraic structures, it studies the use of variables in equations and how to manipulate these equations.

Algebra is often understood as a generalization of arithmetic. Arithmetic studies arithmetic operations, like addition, subtraction, multiplication, and division, in a specific domain of numbers, like the real numbers. Elementary algebra constitutes the first level of abstraction. Like arithmetic, it restricts itself to specific types of numbers and operations. It generalizes these operations by allowing indefinite quantities in the form of variables in addition to numbers. A higher level of abstraction is achieved in abstract algebra, which is not limited to a specific domain and studies different classes of algebraic structures, like groups and rings. These algebraic structures are not restricted to typical arithmetic operations and cover other binary operations besides them. Universal algebra is still more abstract in that it is not limited to binary operations and not interested in specific classes of algebraic structures but investigates the characteristics of algebraic structures in general.

The term "algebra" is sometimes used in a more narrow sense to refer only to elementary algebra or only to abstract algebra. When used as a countable noun, an algebra is a specific type of algebraic structure that involves a vector space equipped with a certain type of binary operation. Depending on the context, "algebra" can also refer to other algebraic structures, like a Lie algebra or an associative algebra.