An algebraic structure, or algebraic system, consists of a nonempty underlying set together with a collection of operations on that set, typically binary operations such as addition and multiplication, and a finite set of identities, called axioms, that those operations must satisfy. An algebraic structure may itself be built from other algebraic structures, as when a vector space involves a second structure called a field together with a scalar multiplication operation between the field's scalars and the vector space's vectors. Abstract algebra is the name commonly given to the study of algebraic structures, formalized generally in universal algebra, while category theory offers another formalization covering structures of the same type along with the structure-preserving maps, called homomorphisms, between them. In universal algebra an algebraic structure is called an algebra, a term that is itself ambiguous since elsewhere an algebra means a structure that is a vector space over a field or a module over a commutative ring.
Facts
Origin YearMarks the first volume of B. L. van der Waerden Moderne Algebra (1930-1931), which this source credits with reorienting algebra from the theory of equations to the theory of algebraic structures; earlier work by Fraenkel (1914) and Noether (1920) on specific structures preceded this synthesis. Connections
Associated With
Sets, Concepts Source Algebraic Structure (Wikipedia)
Attributed To
Source Abstract Algebra (Wikipedia)
Source Abstract Algebra (Wikipedia)
In Branch
Source Algebraic Structure (Wikipedia)
Kind Of Object Of
Sources
1. Abstract Algebra (Wikipedia)
Wikimedia FoundationModern algebra
These developments of the last quarter of the 19th century and the first quarter of the 20th century were systematically exposed in Bartel van der Waerden's Moderne Algebra, the two-volume monograph published in 1930-1931 that reoriented the idea of algebra from the theory of equations to the theory of algebraic structures.
Attributed To: Walther von Dyck, Lead
Walther von Dyck in 1882 was the first to require inverse elements as part of the definition of a group
Attributed To: Richard Dedekind, Lead
Dedekind and Miller independently characterized Hamiltonian groups and introduced the notion of the commutator of two elements
View the Source Algebraic Structure (Wikipedia)
Wikimedia FoundationLead section
In mathematics, an algebraic structure or algebraic system consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities (known as axioms) that these operations must satisfy.
In Branch: Algebra, Lead section, third paragraph
Abstract algebra is the name that is commonly given to the study of algebraic structures.
Associated With: Sets, Lead section, first paragraph
an algebraic structure or algebraic system consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities (known as axioms) that these operations must satisfy.
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