Weak ordering

Remarks

Let $f \colon A \to \R$ be any function and set $a < b$ when $f(a) < f(b)$. This is a strict weak ordering: $a$ and $b$ are incomparable exactly when $f(a) = f(b)$, and equality of values is transitive. On $A = \set{a, b, c}$, the strict partial order whose only relation is $a < b$ is not a strict weak ordering. Here $a$ and $c$ are incomparable and $c$ and $b$ are incomparable, but $a$ and $b$ are comparable. In a strict weak ordering, incomparability is an equivalence relation, and $<$ induces a strict total order on its classes. The non-strict form is a total preorder: a preorder $\lesssim$ in which $a \lesssim b$ or $b \lesssim a$ for all $a, b$. Strict weak orderings and total preorders on $A$ correspond one to one, by $a \lesssim b$ if and only if not $b < a$.