Algebraic structure

An algebraic structure is a set equipped with one or more operations satisfying specified axioms. Groups, rings, and vector spaces are familiar examples.

Set: A collection of distinct objects treated as a whole. The elements on which an algebraic structure's operations are defined.
Group: A set with an associative binary operation, an identity element, and inverses for every element. Groups are the basic example built from one operation and a short list of axioms.
Universal algebra: The study of algebraic structures through operations, identities, and structure-preserving maps. It provides a general framework for describing structures using shared algebraic language.
Magma: A set equipped with a binary operation, with no further laws required. It provides the minimal starting point against which stronger algebraic axioms stand out.
Évariste Galois: A French mathematician whose work connected polynomial equations with permutation groups. Galois's ideas helped establish groups as abstract objects arising across different problems.
Binary operation: A rule that combines two elements of a set to produce an element. Many familiar structures begin with one or more operations of this kind.
Ring: A set with addition and multiplication, where addition forms an abelian group and multiplication distributes over addition. Rings show how axioms coordinate two operations on one set.
Direct product: A construction that combines structures coordinate by coordinate while preserving their operations. It builds larger structures from several structures of the same kind.
Semigroup: A set with an associative binary operation. Associativity alone distinguishes semigroups from structures requiring identities or inverses.
Arthur Cayley: A nineteenth-century British mathematician who developed major parts of modern abstract algebra. Cayley's work helped turn groups from transformations into objects studied through axioms.

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