Preorder

Definition

Let $A$ be a set. A preorder on $A$ is a binary relation $\le$ on $A$ that is reflexive and transitive. For all $a, b, c \in A$:

1. $a \le a$;
2. $a \le b$ and $b \le c$ imply $a \le c$.

Remarks

Divisibility on $\mathbb{Z}$ is a preorder. It is not antisymmetric, because $2 \mid -2$ and $-2 \mid 2$. An antisymmetric preorder is a partial order. For a preorder, $a \le b$ and $b \le a$ together define an equivalence relation. The preorder gives a partial order on the quotient set.

Explanations

L1 human

Compare people by height with "is at most as tall as". This is a preorder: two different people can have the same height.