Preorder

A preorder on a set X is a relation \mathbin{\leqslant} \subseteq X \times X on X that is

1. reflexive: \forall x \in X.\ x \leqslant x
2. transitive: \forall x, y, z \in X.\ (x \leqslant y \land y \leqslant z) \Longrightarrow x \leqslant z

Versus partial orders

A preorder is almost a partial order, but it does not satisfy the antisymmetry axiom. In other words, it is possible to have that two elements are equivalent according to the preorder, in the sense that

x \sim y\quad
\triangleq\quad
  x \leqslant y
    \quad\land\quad
  y \leqslant x

without them being equal (x \neq y).

There is a canonical way to induce a partial order by quotienting with this equivalence relation \sim.

References

• PlanetMath: preorder