Strict Weak Ordering Induces Partition
- $\mathbb S$ is the set of these partitions of $S$;
- $<$ is the strict total ordering on $\mathbb S$ induced by $\prec$.
From the definition of strict weak ordering, we define the symbol $\Bumpeq$ as:
- $a \Bumpeq b := \neg a \prec b \land \neg b \prec a$
that is, $a \Bumpeq b$ means "$a$ and $b$ are non-comparable".
Checking in turn each of the criteria for equivalence:
As $\prec$ is antireflexive, by definition $\forall a \in S: \neg a \prec a$.
Hence by the Rule of Idempotence $\neg a \prec a \land \neg a \prec a$ and so $\forall a \in S: a \Bumpeq a$.
We have that $a \Bumpeq b$ is defined as being $\neg a \prec b \land \neg b \prec a$.
It follows from the Rule of Commutation that $\neg b \prec a \land \neg a \prec b$, and so $b \Bumpeq a$.