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In Mathematical, a external law of composition in a unit E with operators (or scalars) in S (one also more briefly says a external law of S on E ) is a external ternary Relation of S on E which is also a application.

Definition

According to whether S comes in first or in the second place in the Cartesian Produit which is used as starting together with the external law considered, one distinguishes the external laws on the left and on the right. As follows:

a external law on the left of S on E is an application of S × E in E ;
a external law on the right of S on E is an application of E × S in E .

Principal properties

Simple properties

That is to say a unit E provided with an external law “. ” with scalars in a unit S . We will consider the case of a law on the left (resp. on the right).

the law “. ” is exo-unifère on the left (resp. exo-unifère on the right ), or more simply unifère if there exists an element of S which, composed by this law with any element of E , gives again the element of E

or:

- for a relation on the left:

\ exists \ \ epsilon \ in S/\ \ forall \ X \ in E, \ \ epsilon. X = X \,

- and on the right:

\ exists \ \ epsilon \ in S/\ \ forall \ X \ in E, \ X. \ epsilon = X \,

the law “. ” absorbing on the right (resp is . absorbing on the left ), or more simply absorbing if there exists an element of E which, composed by this law with any element of S , gives again itself

or:

- for a relation on the left:
$\ exists \ has \ in E/\ \ forall \ \ lambda \ in S, \ \ lambda. has = has \,$
- and on the right:
$\ exists \ has \ in E/\ \ forall \ \ lambda \ in S, \ A. \ lambda = has \,$

the law “. ” exo-absorbing on the left (resp is . exo-absorbing on the right ), or more simply exo-absorbing if there exists an element of E and an element of S such as the element of E is the single result of the composition of the element of S with any element of E

or:

- for a relation on the left:
$\ exists \ has \ in E, \ exists \ \ Omega \ in S/\ \ forall \ X \ in E, \ \ Omega. X = has \,$
- and on the right:
$\ exists \ has \ in E, \ exists \ \ Omega \ in S/\ \ forall \ X \ in E, \ X. \ Omega = has \,$

the law “. ” is on the left regular (resp. on the right ) if for each element of S , its compounds by this law with the elements of E are all distinct between them

or:

- for a relation on the left:
$\ forall \ \ lambda \ in S, \ forall \ (X, there) \ in E^2, \ lambda. X = \ lambda. there \ \ Rightarrow (X = there) \,$
- and on the right:
$\ forall \ \ lambda \ in S, \ forall \ (X, there) \ in E^2, X. \ lambda = Y. \ lambda \ \ Rightarrow (X = there) \,$

the law “. ” is on the right exo-regular ( resp. on the left ) if for each element of E , its compounds by this law with the elements of S are all distinct between them

or:

- for a relation on the left:
$\ forall \ (\ lambda, \ driven) \ in S^2, \ forall \ X \ in E, \ lambda. X = \ driven. X \ \ Rightarrow (\ lambda = \ driven) \,$
- and on the right:
$\ forall \ (\ lambda, \ driven) \ in S^2, \ forall \ X \ in E, X. \ lambda = X. \ driven \ \ Rightarrow (\ lambda = \ driven) \,$

the law “. ” regular is if it is regular on a side and exo-regular other.

Properties relative to an internal law

the law “. ” is exo-associative compared to a law interns “ $* \,$ ” of S if any compound by the law “. ” of a scalar with the compound by the law “. ” of another scalar and an element of E equal to is composed of this element of E with the compound of the two scalars by the law “ $* \,$ ”

or:

- for a relation on the left:
$\ forall \ (\ lambda, \ driven) \ in S^2, \ forall \ X \ in E, \ \ lambda. (\ driven. X) = (\ lambda * \ driven). X \,$
- and on the right:
$\ forall \ (\ lambda, \ driven) \ in S^2, \ forall \ X \ in E, \ (X. \ driven). \ lambda = X. (\ driven * \ lambda) \,$

the law “. ” is on the left distributive ( (resp. on the right ) ) compared to a law interns “ $\ club-footed \,$ ” of E if any compound by the law “. ” of a scalar with the compound by the law “ $\ club-footed \,$ ” of two elements of E equal to is composed by the law “ $\ club-footed \,$ ” of both composed by the law “. ” of these elements of E with the preceding scalar

or:

- for a relation on the left:
$\ forall \ \ lambda \ in S, \ forall \ (X, there) \ in E^2, \ \ lambda. (X \ club-footed there) = (\ lambda. X) \ club-footed (\ lambda. there) \,$
- and on the right:
$\ forall \ \ lambda \ in S, \ forall \ (X, there) \ in E^2, \ (X \ club-footed there). \ lambda = (X. \ lambda) \ club-footed (Y. \ lambda) \,$

the law “. ” is on the right exo-distributive ( (resp. on the left ) ) compared to a law interns “ $\ signal \,$ ” of S relative with another law interns “ $\ club-footed \,$ ” of E if any compound by the law “. ” of an element of E with the compound by the law “ $\ signal \,$ ” of two scalars equal to is composed by the law “ $\ club-footed \,$ ” of both composed by the law “. ” of the element of E with each scalar

or:

- for a relation on the left:
$\ forall \ (\ lambda, \ driven) \ in S^2, \ forall \ X \ in E, \ (\ lambda \ signal \ driven). X = (\ lambda. X) \ club-footed (\ driven. X) \,$
- and on the right:
$\ forall \ (\ lambda, \ driven) \ in S^2, \ forall \ X \ in E, \ X. (\ lambda \ signal \ driven) = (X. \ lambda) \ club-footed (X. \ driven) \,$

See too

Law of composition

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