obtaining an equivalence relation from an ordering relation

Question

given a set A and a total ordering < on A. can we use this ordering relation to obtain an equivalence relation on A and thus obtaining a partition on A? I was thinking of this: let X = { x belongs to A| x<a} likewise X`= { x belongs to A| x>a} and the set {a} for some chosen a in A these three sets represent a partition on A the thing is that I have a feeling that there is some assumption about (A,<) which I am not aware of and I am using it to get the above partition. what do you think?

Answer 1

If the final goal is to get a partition of $A$, then you can simply say that you will take the partition$$\bigl\{\{x\in A\mid x<a\},\{a\},\{x\in A\mid a<x\}\bigr\}\tag1$$(which is a partition). There is no need to introduce an equivalence relation for that. But if you really want it, you can define it as follows:$$x\sim y\iff x,y<a\vee x=y=a\vee a<x,y.$$Then $\sim$ is an equivalence relation on $A$ and $(1)$ is the set of equivalence classes of $(A,\sim)$. But this is a waste of time, in my opinion.