Relation of order

A relation of order in a Ensemble $E$ is a binary Relation in this unit which makes it possible to compare its elements between them in a coherent way. A unit provided with a relation of order is a together ordered or quite simply a order .

Definitions and examples

Relation of order

A relation of order on a unit E is a binary relation on E reflexive, transitive and antisymmetric

reflexive , if it puts any element in relation to itself, i.e. if:

\ forall X \ in E, \ quad \ X \ mathcal {R} X \,

antisymmetric , if the distinct elements are never in mutual relation, i.e. if:

\ forall (X, there) \ in E^2, \ quad \ X \ mathcal {R} there \; \ wedge \; there \ mathcal {R} X \ Longrightarrow X = there \,

transitive , if two elements are connected as soon as one can forward by a third, i.e.

\ forall (X, there, Z) \ in E^3, \ quad \ X \ mathcal {R} there \; \ wedge \; there \ mathcal {R} Z \ Longrightarrow X \ mathcal {R} Z \,

One can immediately notice that, from the form even of these axioms, they are checked by the opposite or reciprocal relation

\ mathcal {R} ^ {- 1}

, which is defined by

x \ mathcal {R} ^ {- 1} there \ yew there \ mathcal {R} x

With any relation of order is thus associated a reciprocal order (smaller or equal/larger or equal, lower or equal/higher or equal etc).

Examples and counterexamples

the relation “is lower or equal to” is a relation of order on the whole of the entireties (natural or relative), on the whole of rational or the whole of realities. It can be explicitly defined by

\ forall (X, there) \ in E^2, \ quad X \ there the \ yew there - X \ in E_+

where

E_+= \ {X \ in E | X \ geq 0 \}

the relation “is strictly lower than” is not a relation of order because it is not reflexive.
the relation “is multiple of” is a relation of order on the whole of the natural entireties.
the “divided” relation is a relation of order on the natural entireties.
the relation “is multiple of” is not a relation of order on the relative entireties because it is not antisymmetric: 6 is multiple of -6 and -6 is multiple of 6 but 6 is not equal to -6.
the relation “is a Sous-ensemble of” or “is contained in” is a relation of order on the whole of the parts of a unit.
the relation “is placed before” definite on the whole of the point of the trigonometrical circle by

M is before NR if and only if the principal measurement of the angle

(\ overrightarrow {OM}; \ overrightarrow {ONE})

is positive or null is not a relation of order because it is not transitive.

the relation “is the father of” on a whole of people is not a relation of order because it is not reflexive.
the relation defined on the whole of the complexes by

\ forall (X, there) \ in \ mathbb C^2, \ quad X \ there the \ yew there - X \ in \ R_+

is a relation of order.

the relation $\ mathcal R$ defined on the whole of the complexes by

\ forall (X, there) \ in \ mathbb C^2, \ quad X \ mathcal R there \ yew \ mathrm {Re} (X) < \ mathrm {Re} (there) \; \ vee \; (\ mathrm {Re} (X) = \ mathrm {Re} (there) \; \ wedge \; \ mathrm {Im} (X) \ the \ mathrm {Im} (there))

is a relation of order. It is however not compatible with the structure of body of

\ mathbb C

Increasing applications

(E, \ mathcal {R})

and

(F, \ mathcal {S})

are two ordered units, a

f application: E \ to F

is known as increasing if it is compatible with the relations, i.e si :

\ forall (X, there) \ in E^2, \ quad X \ mathcal {R} there \ Longrightarrow F (X) \ mathcal {S} F (there)

Relation of a strict nature

With a relation of order one naturally associates the relation obtained by restricting those with the couples of distinct elements. One then speaks about strict order . If the relation of order is noted

\ le

, one thus defines the relation of a strict nature associated, noted

<

by:

x < there \ yew X \ there the \; \ wedge \; X \ y

One can then specify broad relation of order when one wants to distinguish the concept of relation from order of that of strict order .

It is completely possible to axiomatize the concept of a strict nature directly. That can even prove more natural in certain cases.

A relation of a strict nature is a binary relation irréflexive , and transitive . I.e. a relation $R$ definite on a $E$ unit is a strict order when it checks the two following properties:

(Irreflexivité) no element of $E$ is in relation to itself by $R$ :

\ forall X \ in E, \ quad X \ not \! R x

;

(transitivity) two elements is connected as soon as one can forward by a third, i.e.:

\ forall (X, there, Z) \ in E^3, \ quad \ X \ mathcal {R} there \; \ wedge \; there \ mathcal {R} Z \ Longrightarrow X \ mathcal {R} Z \,

One immediately deduces from these two properties that a relation of a strict nature is antisymmetric. To tell the truth a relation of a strict nature is antisymmetric in a direction stronger than a broad relation of order, i.e. if $x$ is in relation to $y$ by $R$ then $y$ is not in relation to $x$ by this relation. This is why one qualifies sometimes this property of strong antisymetry.

The relation $R$ definite on $E$ is strongly antisymmetric si :

\ forall (X, there) \ in E^2, \ quad X R there \ Longrightarrow there \ not \! R x

However for a irréflexive relation, like the strict orders, this property is equivalent to the property of anti-symmetry defined for the broad orders. There is thus no disadvantage to speak about anti-symmetry in both cases.

With a relation of a strict nature, let us note the $<$ , one associates a relation of a broad nature naturally, note the $\ le$ , defined by:

x \ there the \ yew x

To choose one or the other of axiomatizations does not have importance in oneself. In both cases one defined a broad order and a strict order associated. Indeed it is easily checked, by using the properties of the equality, that:

the relation of a strict nature associated with a relation of a broad nature (transitive, reflexive and antisymmetric) checks the strict axioms of order well (it is transitive and irréflexive).
the broad relation of order associated with a relation of a strict nature (transitive and irréflexive) checks the axioms of a broad nature well (it is transitive, reflexive and antisymmetric).

Complementary properties

Total order, partial order

a broad relation of order is total if it vérifie :

\ forall (X, there) \ in E^2, \ quad X \ there the \; \ vee \; there \ the x

the $E$ unit is then known as completely ordered . It is also said that the relation of order $\ le$ is total or that $(E, \ it)$ is a total order .

a relation of order is partial if it is not total , i.e. if there exist two elements which one can connect, neither in a direction nor in the other:

\ exists (X, there) \ in E^2, X \ not \ Leq there \; \ wedge \; there \ not \ Leq x

the $E$ unit is then known as partially ordered .

Two elements $x$ and $y$ such as ( $x \ there the \; \ vee \; there \ the x$ ) are known as comparable . The comparability is to some extent the symmetrization of a relation of order. Thus a total order is an order of which all the elements are two to two comparable.

Exemples :

the unit $\ R$ provided with the relation of order $\ le$ is completely ordered.
the whole of the natural entireties provided with the relation of divisibility is partially ordered.

For a strict order, totality is expressed ainsi :

$\ forall (X, there) \ in E^2, \ quad X < there \; \ vee \; X = there \; \ vee \; there < x$

and it is said whereas it is a relation of a total strict nature . There is no possible confusion with the preceding direction of total relation, because a relation of a strict nature, which is irréflexive, cannot be total with the direction where is a broad order.

For a total strict order, the three possibilities - $x < there, X = there, there < x$ - are exclusive FIXME, and one speaks sometimes, following Cantor, of property of trichotomy .

Negation (or complementary) of a relation of order

The negation of a binary relation

R

definite on a

A

unit is the relation of graph the Complémentaire of that of

R

A \ times A

. It is noted

\ not \! R

. Known as differently, two elements are in relation by

\ not \! R

if and only if they are not it by

R

To say that an order is total, it is to say that its negation is the opposite strict order. I.e. there is equivalence for an order $\ leq$ between:

$\ leq$ is total;
$x \ not \ Leq there \ yew there < x$

On the other hand as soon as there exist two noncomparable distinct elements by an order, its negation cannot be an order (strict or broad), because it is not antisymmetric. The negation of a nontotal order is thus never an order.

For example, the negation of inclusion $\ not \ subset$ on the whole of the parts of a whole of at least two elements is not an order, because, if $a \ not= b$ , one has $\ {has \} \ not \ subset \ {B \}$ and $\ {B \} \ not \ subset \ {has \}$ .

Compatibility

A relation of order

\ le

on a

E

unit provided with a Law of composition interns

*

is compatible with this law if and only si :

$\ forall (X, there, Z) \ in E^3, \ quad X \ there the \ Longrightarrow X * Z \ there * Z \; \ wedge \; Z * X \ Z * y$

the relation of order $\ le$ on the whole of realities is compatible with the addition but not with the multiplication.
the relation of order $\ le$ on the whole of strictly positive realities is compatible with the addition and the multiplication.
the unit $\ mathbb {C}$ of the complex numbers is not ordonnable by a relation between order compatible and the operations of addition and multiplication.

If a group is provided with a relation between order compatible and its internal law, it is called Groupe ordered.

A completely ordered group checking the property

\ forall (X, there) \ in (G_+) ^2, \ quad \ exists N \ in \ mathbb NR, \ quad X \ N y

Archimédien is known as.

Ordered well together

An ordered unit is known as ordered well if all left not vacuum this unit has a smaller element.

Lattice

A unit is called lattice if it is ordered and that any couple of elements has a higher Borne and a lower Borne.

Diagram of Hasse

When one works on a finished order, it can be pleasant to have a visual representation of this one. One can propose of them one which is similar to the usual representation of a graph on paper. It is the Diagramme of Hasse .

Derived concepts

Pre-order

A Pre-order is a binary relation reflexive and transitive .

It would be a relation of order in which one would authorize the noncommonplace cycles (i.e. cycles of more than one element). To add anti-symmetry makes impossible the presence of these cycles noncommonplace.

See too

comparability

Relation of equivalence
Lattice
complete Order partial
binary Relation
Diagram of Hasse
Group ordered
Body ordered

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