Binary operation

In mathematics, a binary operation or dyadic operation is a rule for combining two elements, called operands, to produce a third element. More formally, a binary operation on a set is a function that maps pairs of elements from the set to a single element within the same set, a property known as closure. Binary operations are the foundational building blocks of algebraic structures, providing the essential mechanisms through which elements interact in fields such as abstract algebra, logic, and computer science.

Formal Definition

Let $S$ be a set. A binary operation on $S$ is a function $f: S \times S \to S$. Because the domain of $f$ is the Cartesian product $S \times S$, the operation takes exactly two arguments from $S$. The codomain being $S$ ensures that the operation is closed; that is, applying the operation to any two elements of $S$ always yields another element of $S$. It is common to denote a binary operation by a symbol such as $$, $\circ$, $\cdot$, or $+$, writing $f(a, b)$ as $a b$.

If the operation is defined on two different sets, say $f: A \times B \to C$, it is sometimes referred to as a general binary operation, though in the context of algebraic structures, the term usually implies a single underlying set. When the operation involves elements from a set and an external set (such as scalar multiplication of a vector space, $f: F \times V \to V$), it is specifically called an external binary operation or a binary action.

Properties of Binary Operations

Binary operations are often classified and studied based on the algebraic properties they satisfy. The most significant properties include:

• Commutativity: A binary operation $$ on a set $S$ is commutative if $a b = b * a$ for all $a, b \in S$. Addition of real numbers is commutative, whereas subtraction and matrix multiplication are not.
• Associativity: An operation $$ is associative if $(a b) c = a (b * c)$ for all $a, b, c \in S$. Associativity allows expressions to be evaluated without strict reliance on parentheses. Both addition and multiplication of real numbers are associative.
• Identity Element: An element $e \in S$ is an identity element for $$ if $a e = e * a = a$ for all $a \in S$. For addition of integers, the identity is 0; for multiplication, it is 1.
• Inverse Element: Given an identity element $e$, an element $b \in S$ is an inverse of $a \in S$ if $a b = b a = e$. In the real numbers under addition, the inverse of $a$ is $-a$.
• Idempotence: An operation is idempotent if $a * a = a$ for all $a \in S$. The maximum and minimum functions on real numbers, as well as the union and intersection operations on sets, are idempotent.
• Distributivity: This property involves two binary operations, say $$ and $\circ$. The operation $$ distributes over $\circ$ if $a (b \circ c) = (a b) \circ (a c)$ (left distributivity) and $(b \circ c) a = (b a) \circ (c a)$ (right distributivity) for all $a, b, c \in S$. Multiplication distributes over addition in the real numbers.

Partial Binary Operations

A partial binary operation on a set $S$ is a function $f: D \to S$, where $D$ is a proper subset of $S \times S$. This means the operation is not defined for all possible pairs of elements in $S$. A classic example is division on the set of real numbers $\mathbb{R}$. Division is a partial binary operation because dividing by zero is undefined; thus, the domain of the division function excludes all pairs $(a, 0)$ where $a \in \mathbb{R}$. Similarly, certain geometric constructions and the subtraction of natural numbers (where the result must remain a natural number) form partial binary operations.

Examples in Mathematics

Binary operations are ubiquitous across various branches of mathematics:

• Arithmetic: The basic operations of addition, subtraction, multiplication, and division (as a partial operation) on the sets of natural numbers, integers, rational numbers, real numbers, and complex numbers.
• Set Theory: Union, intersection, symmetric difference, and set difference are binary operations on the power set of any given set.
• Logic: Logical connectives such as conjunction (AND), disjunction (OR), exclusive disjunction (XOR), implication, and biconditional are binary operations on the set of truth values.
• Linear Algebra: Matrix addition and matrix multiplication are binary operations on the set of matrices of compatible dimensions. The dot product and cross product are binary operations on vectors.
• Function Composition: The composition of functions is a binary operation on the set of all functions from a set to itself.

Role in Abstract Algebra

In abstract algebra, binary operations are the primary tool used to define algebraic structures. The specific properties that a binary operation satisfies determine the type of structure formed by the set and the operation:

• Magma: A set equipped with a single closed binary operation. No other properties are required.
• Semigroup: A magma where the binary operation is associative.
• Monoid: A semigroup that possesses an identity element.
• Group: A monoid where every element has an inverse element. Groups are central to the study of symmetry in mathematics and physics.
• Ring: A set equipped with two binary operations (usually called addition and multiplication), where the set forms an abelian (commutative) group under addition, a monoid under multiplication, and multiplication distributes over addition.
• Field: A ring where the non-zero elements form an abelian group under multiplication.
• Lattice: A set equipped with two commutative, associative, and idempotent binary operations (meet and join) that satisfy the absorption laws.

Notation and Syntax

The way binary operations are written varies by context and discipline. The most common notations include:

• Infix Notation: The operator is placed between the operands (e.g., $a + b$ or $x \times y$). This is the standard convention in elementary arithmetic and algebra.
• Prefix Notation (Polish Notation): The operator precedes the operands (e.g., $+ a b$). This notation eliminates the need for parentheses and is frequently used in logic and computer science, notably in Lisp programming languages.
• Postfix Notation (Reverse Polish Notation): The operator follows the operands (e.g., $a b +$). Like prefix notation, it requires no parentheses and is used in stack-based programming languages and some calculators.
• Juxtaposition: In many algebraic contexts, particularly when denoting multiplication in groups or rings, the operation symbol is omitted entirely, and the operands are placed adjacent to each other (e.g., $ab$ instead of $a \cdot b$).