Binary operation

An algebraic operation on a set $A$ with two operands in a given order, hence a function from $A\times A \rightarrow A$. Such an operator may be written in conventional function or prefix form, as $f(a,b)$, occasionally in postfix form, as $a\,b\,\omega$ or $(a,b)\omega$, but more commonly in infix form as $a \star b$ where $\star$ is the operator symbol. Many arithmetic, algebraic and logical functions are expressed as binary operations, such as addition, subtraction, multiplication and division of various classes of numbers; conjunction, disjunction and implication of propositions.

A binary operation is partial if it is not defined on all pairs $(a,b) \in A \times A$ (as for example division by zero is not defined). Properties of binary operations which occur in many contexts include

• Commutativity: $a \star b = b \star a$;
• Associativity: $a \star (b \star c) = (a \star b) \star c$;
• Idempotence: $a \star a = a$.

References

[a1]	R.H. Bruck, "A survey of binary systems" Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. 20 Springer (1958) Zbl 0081.01704