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Algebraic operation

The notion of the operation (especially that of the binary operation) is the keystone of the algebra.

Let be a non-empty set and a non-negative integer.¹ Let , times for and . Under an -ary operation (or shortly an operation) on we understand a function . This means that an -ary operation assigns to every ordered -tuple of elements of a unique element from .

The number is called the arity or the rank of the operation and the inputs of the underlying function are called operands.

Sometimes it is useful to consider a constant as an operation of arity 0, such operation is called nullary. Thus

• If the operation is called nullary. A nullary operation actually specifies a unique element. Put another way, a set with nullary operation is a pointed set with an element distinguished be the operation.. For instance, the existence of the identity element  is a nullary operation.
• If the operation is called unary, and this is a function . Thus unary operation is a function with exactly one operand, for instance , where is the inverse element of is a unary operation.
• If the operation is called binary and is usually denoted as . Binary operation is also called a law of composition.
• If the operation is called ternary. It takes three arguments, e.g. triple product in vector analysis is ternary.
• If the operation is quaternary means; quinary means , sextary means 6-ary, etc.
• Polyadic (or multary) means any number of operands.

A non-empty subset of is called closed with respect to the operation defined on if the restriction is an operation on .

Binary relation on a set is called compatible with an  -ary operation defined on if for all such that , .

A non-empty set endowed with one or more operations is called algebraic structure or algebra. More precisely,  an (universal) algebra is a pair , in which is a non-empty set, and is a set operations on .  

Let and be two non-empty sets endowed with -ary operations and , respectively, where is a non-negative integer.  A map is called a homomorphism (with respect to operations and ) if

In words, if    maps the result under the operation to the result of -maps under the operation .  If is one-to-one (injective) and onto (surjective), that is,  is a bijection, then is called an isomorphism.

Lemma: Let and be two non-empty sets endowed with -ary operations and , respectively, where is a non-negative integer and map a bijection. If is a homomorphism, then so its inverse .

Let , is a system of non-empty sets each with an -ary operation , then we can define (the so-called componentwise definition) a new operation on the Cartesian product by

In many cases it is useful to drop the assumption that is a function with a domain , that is that   is defined for ordered -tuples of elements of  .  If is defined only on a subset of then is called a partial -ary operation.  Typical example of a partial binary operation is the division of real numbers where division by   is not defined.

Notes

1	In general can be any ordinal number.

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