binaryoperation

A binary operation on a set S is a rule that combines any two elements a and b in S to produce a third element a*b that also lies in S. Formally it is a function *: S × S → S. The term binary reflects that the operation takes two inputs. In most mathematical contexts the operation is assumed to be total, meaning it is defined for every pair (a, b) in S × S; some discussions allow partial binary operations where the result is undefined for some pairs.

Key properties often discussed for binary operations include closure (the result a*b remains in S), associativity (a*(b*c) = (a*b)*c for all a, b, c in S), and commutativity (a*b = b*a for all a, b in S). An identity element e satisfies a*e = e*a = a for all a in S. If every element has an inverse with respect to the operation, the structure is a group; if associativity and identity are present, it is a monoid. When two binary operations are present and one distributes over the other, the structure can become a ring or a field, depending on additional axioms.

Examples include addition on the integers, multiplication on the real numbers, string concatenation on the set of finite strings, logical conjunction on booleans, and set union on a power set. In linear algebra, vector addition is a binary operation on a vector space.

Binary operations are foundational in algebra, describing how elements combine and enabling the study of algebraic structures such as semigroups, monoids, groups, rings, and fields. They also model many computational and logical processes that take two inputs and produce a single output.