An algebraic structure is a set that obeys a short lawbook
Take a nonempty set, add operations such as addition or multiplication, and insist on a finite list of identities. That package—the carrier set, the operations, and the axioms—is an algebraic structure, the raw material of abstract algebra.
The underlying set is also called the carrier or domain. Operations are usually binary, though structures may include unary maps, nullary constants, or higher-arity combinations. Some structures sit on top of others: a vector space needs a field of scalars and a scalar-multiplication operation linking the two. Abstract algebra is the study of such packages; universal algebra formalizes the general theory, while category theory tracks structures together with the homomorphisms between them.
In universal algebra a structure is often called an algebra—a word that elsewhere means a vector space or module with a bilinear product—so context matters. The class of all structures with the same operations and the same identity laws is a variety in the universal-algebra sense, not to be confused with an algebraic variety in geometry. In category theory, those structures plus their homomorphisms form a concrete category. The payoff is reuse: once a theorem uses only the axioms, it applies to every system that satisfies them.
Familiar arithmetic laws—associativity, commutativity—need not all hold. Rigid motions in three-space compose associatively but not commutatively. Axioms are often identities that remain true for every substitution of elements. Existential clauses such as "there exists an inverse" can be rewritten by adding auxiliary operations—an identity constant of arity zero, or a unary inversion map—turning the claim into an identity, which is what varieties in universal algebra allow. A group is the package with one associative binary operation, an identity, and inverses for every element. Fields need an inversion axiom for nonzero elements that cannot be reduced to pure identities, so fields do not form a universal-algebra variety. That boundary shows how much geometry of ideas is encoded in the shape of the axioms themselves.
Source: Algebraic structure
