Concept

Algebraic Structure

Algebra

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 Year

1930 1

Marks 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.

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Sets

Source Algebraic Structure (Wikipedia)

In Branch

Algebra

Source Algebraic Structure (Wikipedia)

Kind Of Object Of

Algebraic Element

Entity-backed identity for the object-kind enum value this mathematical object already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The object-kind fact itself stays on the object unchanged.

Alternating Group

Braid Group

Complexification (Lie Group)

Mathematical Structure

Sources

1. Abstract Algebra (Wikipedia)

Wikimedia FoundationHistory section

Quote, History section

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 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.

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Algebraic Structure (Wikipedia)

Wikimedia Foundation

Lead 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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