Pre-order

A pre-order is a binary Relation reflexive and transitive .

I.e., if E is a Ensemble, then ${\ mathcal R} \ subseteq E \ times E$ is an pre-order if and only if:

$\ forall X \ in E \ implies (X, X) \ in {\ mathcal R}$ (reflexivity)
$\ forall X \ forall there \ forall Z {\ mathcal R} \ and (there, Z) \ in {\ mathcal R} \ implies (X, Z) \ in {\ mathcal R}$ (transitivity)

An antisymmetric pre-order is a order.

A symmetrical pre-order is a Relation of equivalence.

Example

On the tops of a graph directed, the relation “being accessible since” is an pre-order (it is in fact the closing reflexive and transitive of the graph). If the graph is without cycle, this relation becomes an order.

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