Suppose you have a set with an equivalence relation $\equiv $ which is not the same as identity and a preorder $\leq$ such that
reflexive equiv $a\equiv a$
symmetric equiv $a\equiv b\rightarrow b\equiv a$
transitive equiv $a\equiv b\wedge b\equiv c\rightarrow a\equiv c$
reflexive leq $a\leq a$
pseud-antisymmetric leq $a\leq b\wedge b\leq a\rightarrow a\equiv b$
transitive leq $a\leq b\wedge b\leq c\rightarrow a\leq c$
There are, however, many pairs $a,b$ such that $a\leq b\wedge b\leq a\wedge a\neq b$.
How would this normally be treated? Would you call $\leq$ a partial order and just ignore the difference between $\equiv$ and $=$ or would you call this something else?
Also, I want this structure to impose a total order on the equivalence classes of $\equiv$. That is, let $x,y,z$ represent equivalences classes of $\equiv$ and let $x\leq y$ be defined as $\exists a\in x,b\in y.a\leq b$, then
$$x\leq x$$
$$x\leq y\wedge y\leq x\rightarrow x=y$$
$$x\leq y\wedge y\leq z\rightarrow x\leq z$$
$$x\leq y\vee y\leq x$$
From intuitions about the model I have in mind (geometric magnitudes) I believe that it should be possible to prove that this structure exists, but it isn't obvious to me how to do that.