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An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field , and an operation called scalar multiplication between elements of the field (called scalars ), and elements of the vector space (called vectors ).
In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a ...
For example, the tensor algebra construction on a vector space is the left adjoint to the functor on associative algebras that ignores the algebra structure. It is therefore often also called a free algebra. Likewise the symmetric algebra and exterior algebra are free symmetric and anti-symmetric algebras on a vector space.
The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether. [36] Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics.
In algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties. The notion of algebraic structure has been formalized in universal algebra
Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a group and the pair of which together make it into a field. A measure: intervals of the real line have a specific length , which can be extended to the Lebesgue measure on many of its subsets .
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