Universal algebra
= Universal algebra
Universal algebra studies <algebraic structures> by specifying their operations and identities in a common language. A signature assigns an <arity> to each operation symbol. A structure interprets each symbol as a <function> $A^n\to A$ on its underlying <set>; an identity requires two terms to agree for every assignment of their variables.
For example, a <group> can be presented with a <binary operation>, a <unary operation> for inversion, and a <nullary operation> for its identity element. A <ring> adds multiplication and its distributive identities. These signatures differ, but the same definitions of a structure-preserving <homomorphism>, a compatible equivalence relation, and a <quotient structure> apply to both.
This shared framework explains why many proofs recur across algebra: a <homomorphism> identifies elements in a way compatible with every specified operation. The resulting quotient must preserve all those operations, not only a selected multiplication table. Universal algebra isolates that compatibility before considering special properties of <groups>, <rings> or other examples.