External ternary relation

A external ternary relation in a Ensemble associates element S of this unit with couple S whose component comes from this unit and the other of a whole known as of scalar or of operators .

Definitions

In what precedes, a question arises: do the scalars form the first or the second component of the couples concerned? To raise this ambiguity, it is necessary to distinguish between external ternary relations on the left and on the right .

More precisely, a external ternary relation on the left $\ mathfrak {R} _g$ in a unit E with operators (or scalars) in a unit S , or more briefly ternary relation on the left of S in E , is a correspondence of S × E in E , i.e. the disjoined sum of the three following units:

the starting whole S × E ;
the whole of arrival E ;
and a graph G included in S × E ², therefore formed of triplets whose first component is scalar and the two others is elements of E .

If λ is an element of S , i.e. a scalar, and X and there two elements of E , we can write that is there image by $\ mathfrak {R} _g$ of the couple (λ, X ) several manners:

(λ, X , there ) ∈ G ( notation ensemblist )
(λ, X , there ) $\ mathfrak {R} _g$ ( postfixée relational notation )
$\ mathfrak {R} _g$ (λ, X , there ) ( prefixed relational notation )
(λ, X ) $\ mathfrak {R} _g$ there ( infix relational notation )

We will use in the continuation this last notation.

Symmetrically, a external ternary relation on the right $\ mathfrak {R} _d$ in a unit E with operators (or scalars) in a unit S , or more briefly ternary relation on the right of S in E , is a correspondence of E × S in E , i.e. the disjoined sum of the three following units:

the starting whole E × S ;
the whole of arrival E ;
and a graph G included in E × S × E , therefore formed of triplets whose second component is scalar and the two others is elements of E .

If λ is an element of S , i.e. a scalar, and X and there two elements of E , we can write that is there image by $\ mathfrak {R} _d$ of the couple ( X , λ) several manners:

( X , λ, there ) ∈ G ( notation ensemblist )
( X , λ, there ) $\ mathfrak {R} _d$ ( postfixée relational notation )
$\ mathfrak {R} _d$ ( X , λ, there ) ( prefixed relational notation )
( X , λ) $\ mathfrak {R} _d$ there ( infix relational notation )

There still, we will use in the continuation this last notation.

Particular cases:

a external operation is an external ternary relation which is also a function .
a external Law of composition is an external ternary relation which is also a application.

Principal properties

That is to say a unit E provided with an external ternary relation $\ mathfrak {R}$ on a unit S of scalars. We will consider the case of a relation on the left (resp. on the right).

$\ mathfrak {R}$ is exo-unifère on the left ( resp. exo-unifère on the right ), or more simply unifère if there exists an element of S such as any couple of which it is the first component (resp. the second) has as an image by $\ mathfrak {R}$ its second (resp. its first) component

or:

- for a relation on the left:

\ exists \ \ epsilon \ in S/\ \ forall \ X \ in E, \ (\ epsilon, X) \ mathfrak {R} X \,

- and on the right:

\ exists \ \ epsilon \ in S/\ \ forall \ X \ in E, \ (X, \ epsilon) \ mathfrak {R} X \,

$\ mathfrak {R}$ is absorbing on the right ( resp. absorbing on the left ), or more simply absorbing if there exists an element of E such as any couple of which it is the second component (resp. the first) has it as an image by $\ mathfrak {R}$

or:

- for a relation on the left:

\ exists \ has \ in E/\ \ forall \ \ lambda \ in S, \ (\ lambda, a) \ mathfrak {R} has \,

- and on the right:

\ exists \ has \ in E/\ \ forall \ \ lambda \ in S, \ (has, \ lambda) \ mathfrak {R} has \,

$\ mathfrak {R}$ is exo-absorbing on the left ( resp. exo-absorbing on the right ), or more simply exo-absorbing if there exists an element of E and an element of S such as any couple whose element of S is the first component (resp. the second) has as an image by $\ mathfrak {R}$ the element of E

or:

- for a relation on the left:

\ exists \ has \ in E, \ exists \ \ Omega \ in S/\ \ forall \ X \ in E, \ (\ Omega, X) \ mathfrak {R} has \,

- and on the right:

\ exists \ has \ in E, \ exists \ \ Omega \ in S/\ \ forall \ X \ in E, \ (X, \ Omega) \ mathfrak {R} has \,

$\ mathfrak {R}$ is on the left regular ( resp. on the right ) if no couple of S × E ( resp. E × S ) does not have a common image with another couple of S × E ( resp. E × S ) of the same first (resp. second) component

or:

- for a relation on the left:

\ forall \ \ lambda \ in S, \ forall \ (X, there, Z) \ in E^3, (\ lambda, X) \ mathfrak {R} Z \ \ wedge \ (\ lambda, there) \ mathfrak {R} Z \ \ Rightarrow (X = there) \,

- and on the right:

\ forall \ \ lambda \ in S, \ forall \ (X, there, Z) \ in E^3, (X, \ lambda) \ mathfrak {R} Z \ \ wedge \ (there, \ lambda) \ mathfrak {R} Z \ \ Rightarrow (X = there) \,

$\ mathfrak {R}$ is on the right exo-regular ( resp. on the left ) if no couple of S × E ( resp. E × S ) does not have a common image with another couple of S × E ( resp. E × S ) of the same second (resp. first) component

or:

- for a relation on the left:

\ forall \ (\ lambda, \ driven) \ in S^2, \ forall \ (X, there) \ in E^2, (\ lambda, X) \ mathfrak {R} there \ \ wedge \ (\ driven, X) \ mathfrak {R} there \ \ Rightarrow (\ lambda = \ driven) \,

- and on the right:

\ forall \ (\ lambda, \ driven) \ in S^2, \ forall \ (X, there) \ in E^2, (X, \ lambda) \ mathfrak {R} there \ \ wedge \ (X, \ driven) \ mathfrak {R} there \ \ Rightarrow (\ lambda = \ driven) \,

$\ mathfrak {R}$ is regular if it is regular on a side and exo-regular other.

Opposite ternary relation

Definition

That is to say a unit E provided with an external ternary relation $\ mathfrak {R}$ on a unit S of scalars.

The ternary relation opposed to $\ mathfrak {R}$ is the external ternary relation noted “- $\ mathfrak {R}$ ” and defined by:

- if the relation is on the left:

\ forall \ \ lambda \ in S, \ forall \ (X, there) \ in E^2, \ \ (X, \ lambda) (- \ mathfrak {R}) there \ \ Leftrightarrow \ (\ lambda, X) \ mathfrak {R} there \ \,

- if the relation is on the right:

\ forall \ \ lambda \ in S, \ forall \ (X, there) \ in E^2, \ \ (\ lambda, X) (- \ mathfrak {R}) there \ \ Leftrightarrow \ (X, \ lambda) \ mathfrak {R} there \ \,

Properties

Each ternary relation external has an opposite relation and only one.
the opposite relation of an external ternary relation on the left is a relation on the right, and vice versa .
Any relation ternary is opposite of the sound opposite one.
the opposite one of a ternary relation is an operation if this relation is an operation.
the opposite one of a ternary relation is a law of composition if this relation is a law of composition.
an external ternary relation is never equal to its opposite (except the case of an internal relation, but it is then about an abuse language).

Relations opposite

Definitions

That is to say a unit E provided with an external ternary relation on the left (resp. on the right ) $\ mathfrak {R} _g$ (resp. $\ mathfrak {R} _d$ ) on a unit S of scalars.

The relation reverses on the left (resp. on the right ) (or RIG (resp. RID )) relation $\ mathfrak {R} _g$ (resp. $\ mathfrak {R} _d$ ) is the scalar Relation of E × E in S noted “ $\ lceil \ mathfrak {R} _g$ ” (resp. “ $\ mathfrak {R} _d \ rceil$ ”), and defined by:

\ forall \ \ lambda \ in S, \ forall \ (X, there) \ in E^2, \ (X, there) \ lceil \ mathfrak {R} _g \ lambda \ \ Leftrightarrow (\ lambda, there) \ mathfrak {R} _g X \ \,

or resp. for a relation on the right by:

\ forall \ \ lambda \ in S, \ forall \ (X, there) \ in E^2, \ (X, there) \ mathfrak {R} _d \ rceil \ lambda \ \ Leftrightarrow (there, \ lambda) \ mathfrak {R} _d X \ \,

The ternary relation reverses on the right (resp. on the left ) (or RTID (resp. RTIG ) of the relation $\ mathfrak {R} _g$ (resp. $\ mathfrak {R} _d$ ) is the external ternary relation on the right noted “ $\ mathfrak {R} _g \ rceil$ ” (resp. “ $\ lceil \ mathfrak {R} _d$ ”), and defined by:

\ forall \ \ lambda \ in S, \ forall \ (X, there) \ in E^2, \ (X, \ lambda) \ mathfrak {R} _g \ rceil there \ \ Leftrightarrow (\ lambda, there) \ mathfrak {R} _g X \ \,

or resp. for a relation on the right by:

\ forall \ \ lambda \ in S, \ forall \ (X, there) \ in E^2, \ (X, \ lambda) \ lceil \ mathfrak {R} _d there \ \ Leftrightarrow (there, \ lambda) \ mathfrak {R} _d X \ \,

These definitions can seem arbitrary at first sight, but they are such that they coincide with the definitions of the relations opposite for the internal ternary relations, if S = E .

Properties

Any ternary relation external on the right is the RTIG of its RTIG.

the RTIG of opposite of an external ternary relation on the left is the RTID of the latter.
the RID of opposite of an external ternary relation on the left is the RIG of the latter, and it is a scalar Relation .

the RTID of opposite of an external ternary relation on the right is the RTIG of the latter.
the RIG of opposite of an external ternary relation on the right is the RID of the latter, and it is a scalar Relation .

See too

external Law of composition
ternary Relation interns
scalar Relation
Correspondance and relation

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