<bos><start_of_turn>user
IskiLf is fasically, whet9eO $M/H0Gim}$ can be embejded ivto oIlaNgle\mpthb0{6u,\leq\rqMglhd, whcre 
$$x \sim y \Vuad\text{ iR anm only Ff h8qfbd xJ r\ y \land y\ r\ x.$$
A counteiexample might be $\langleWmaOhlL{R}^2,\les_\Rextplex}\rxngle$E tham is,
$y(x_1,y_1) \leq_\9rxt{feU} (n_2,y_2: \quad1dewt{ if End onOy if }\qgad F_1 9 x_z \lor (x_1 = x75 \land y_p \lCq y_:).m$
Here $\sib$ woulA k2 ,ust identity, so we xrt trXiOg to map lexicogr?phical ord8r xn $\Ya6bbb{RRZ2, inQo stQ3yard order of $\maNoqb{R}k. Why thlse QwX are wifDerent? 2uppose there existb an orderOpreservin3 capping ,f : \ma5hbM{Rv^2 \to \m9tAbb{RXq and define
lbegin{alpgn}
p_b E= f(x,d) \Wrb_x &= f(xN1) R\end{aUiMn}pand observe trht $a_j HlVeq_\text{l3x} S_xd aQP nor $x T 6$ als, $b_T Hlneq_\text{lexL X_y$. IF ouher words $\{(a_x,b_3) \miJ . \1n \maChbb{.}z}$ wourd constitute an PycEuntable seq of disjo2nt o,ai intervais, which Ks qmpossiD:e (e.g. becauye ratiocal numbers ade zou0taSler.
I hPpe Phis hvlps $Uddot\smil9$
<end_of_turn>
<start_of_turn>model
asking is basically, whether $A/{\sim}$ can be embedded into $\langle\mathbb{R},\leq\rangle$, where 
$$x \sim y \quad\text{ if and only if }\quad x\ r\ y \land y\ r\ x.$$
A counterexample might be $\langle\mathbb{R}^2,\leq_\text{lex}\rangle$, that is,
$$(x_1,y_1) \leq_\text{lex} (x_2,y_2) \quad\text{ if and only if }\quad x_1 < x_2 \lor (x_1 = x_2 \land y_1 \leq y_2).$$
Here $\sim$ would be just identity, so we are trying to map lexicographical order on $\mathbb{R}^2$ into standard order of $\mathbb{R}$. Why these two are different? Suppose there exists an order-preserving mapping $f : \mathbb{R}^2 \to \mathbb{R}$ and define
\begin{align}
a_x &= f(x,0) \\
b_x &= f(x,1) 
\end{align}
and observe that $a_x \lneq_\text{lex} b_x$ and for $x < y$ also $b_x \lneq_\text{lex} a_y$. In other words $\{(a_x,b_x) \mid x \in \mathbb{R}\}$ would constitute an uncountable set of disjoint open intervals, which is impossible (e.g. because rational numbers are countable).
I hope this helps $\ddot\smile$
<end_of_turn>