Law of composition

In Mathematical, a law of composition , or law very short, is a ternary relation which is also a application . It is thus an application of a Cartesian Produit of two units E and F in a third unit G , with G equal to E or F .

When we define on a unit E a finished number of laws of composition checking certain conditions, we provide the unit with a algebraic Structure . The conditions checked by the laws are called the axioms of the structure of E .

Concept of law

A law (of composition) * : E × F → G , with G = E or G = F , is a application of E × F in G which associates with each couple ( X , there ) of E × F , an element of G usually noted “ X * there ” (instead of the notation functional “* ( X , there )”) and called made up of X and of there , or produced of X and there .

X and is sometimes described there as operands , because a law is only one particular case of operation.

G must be equal to E or F . More precisely:

if E = F = G , the law *: E × E → E is called Law of composition interns in E ;
if E ≠ F and G = F , the law *: E × F → F is called external law of composition on the left on F or external Law of composition , and E is then the field of the operators ;
if E ≠ F and G = E , the law *: E × F → E is called external law of composition on the right on E of field F .

Notice

There exist several notations for the laws:

most current is the notation Infixe ; it is more “speaking”, but requires the recourse to brackets to specify the order of execution of the operations, if there are several of them:

X * there \,

an alternative is the notation by juxtaposition , where the symbol of the law is omitted:

X there \,

the notation Prefix , or Polish , does without brackets:

* X there \,

, sometimes

* X, there \,

the notation Suffix , or Polish reverse , also does without brackets:

X there * \,

, sometimes

X, there * \,

the notation Losange , but requires the recourse to brackets to specify the order of execution of the operations, if there are several of them:

X <> there \,

Examples

a scalar product on a $\ mathbb {K}$ -espace vectorial E is a law of E × E in $\ mathbb {K}$ .
the whole Exponentiation of realities is a law of $\ mathbb {R} \ times \ mathbb {NR}$ in $\ mathbb {R}$ ;
the most current examples of laws of composition are the arithmetic operations, like the Addition, the Soustraction, the Multiplication and the Division; attention however, they are not always laws of composition: thus, the subtraction is not a law of composition in $\ mathbb {NR}$ ;
an external example of multiplication is the multiplication of a vector by a scalar in Linear algebra.

Internal laws

The internal laws are the keystone of the algebraic structures studied in general Algèbre ; they define the groups, the monoids, the semigroups, the rings, etc

The general structure of magma is a unit provided with an unspecified internal law of composition. Many internal laws are commutative or associative, and often have a neutral element and elements symétrisables. The typical examples such laws are the Addition (noted +) and the Multiplication (noted ×) of the numbers or the matrices and also the composition of applications of a whole in itself. However, the multiplication of the matrices or the composition of the applications is not in general commutative.

Examples of laws which are never commutative are the Soustraction (noted -) or the Division (noted ÷ or:).

External laws

Compared to an internal law, an external law utilizes elements of outside, called scalar operators or . An external law E × F → F can be seen like a operation of E on F and one says that E operates on F .

See too

Law of composition interns
external Law of composition
algebraic Structure
universal Algèbre
general Algèbre

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