Binary operation

A binary operation on a set assigns every ordered pair of elements of the set to an element of that same set. It is a function from the set’s Cartesian square to the set.

Function: A relation that assigns each input exactly one output. A binary operation is a function whose inputs are ordered pairs.
Associativity: A property allowing parentheses to be regrouped in repeated applications of an operation without changing the result. It determines whether chains of this operation can be evaluated without specifying parentheses.
Addition: The arithmetic operation that combines quantities to produce their sum. It is a standard binary operation on sets such as the integers and real numbers.
Magma (algebra): A set equipped with a binary operation, with no further axioms required. A magma is the minimal algebraic structure built from the operation alone.
Unary operation: A rule assigning an output to each single element of a set. It contrasts with a binary operation by taking one input rather than a pair.
Cartesian product: The set of all ordered tuples formed from elements of specified sets. The operation’s domain is the set paired with itself.
Commutativity: A property stating that swapping the inputs of an operation leaves its output unchanged. It distinguishes operations where input order matters from those where it does not.
Multiplication: The arithmetic operation that produces a product from two quantities. Its laws alongside addition illustrate how algebra studies multiple operations on one set.
Semigroup: A set with an associative binary operation. Associativity adds the defining extra law beyond a bare operation.
Partial binary operation: A binary operation defined only for some pairs of elements of a set. It differs from the total operations considered here by allowing input pairs without outputs.

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