How to strengthen a graded poset so that items of higher rank always preferred to items of lower rank? Consider a graded poset (partially-ordered set) $(P,\succ)$ with rank function $\rho$.
Does the definition of a graded poset imply that $x \succ y$ for any $x,y\in P$ satisfying $\rho(x) > \rho(y)$?
The definition appears to be stated only in the other direction (if $x \succ y$ then $\rho(x) > \rho(y)$), and the discussion in this question has me question whether it extends to "$x \succ y$ if and only if $\rho(x) > \rho(y)$".
EDIT: The answer to the above is negative. My question then is: What concept/definition would guarantee this property?
 A: Let $P$ be a poset and $\rho:P\to \mathbb N$ be a rank function, that is,
and
$$p\prec q \implies \rho(p)+1=\rho(q),$$
where $p\prec q$ iff $p<q$ and $p<r\leq q$ implies $r=q$ (there is no element between $p$ and $q$).
It follows that
$$p\leq q \implies \rho(p) \leq \rho(q).$$
Let $N=\rho(P) \subseteq \mathbb N$, i.e., $N$ is the set of ranks attributed to elements of $P$.
You asked under what conditions do we have the stronger condition
$$p\leq q \iff \rho(p) \leq \rho(q).$$
The above condition tell us that $\rho$ is an order-embedding, and consequently (see section "Properties" in the linked page), $P\cong N \leq \mathbb N$, that is, up to isomorphism, $P$ is a subset of $\mathbb N$ with the usual ordering, and as such, it's a chain.

Edit.
The above argument works if the requirement were
$$p\leq q \iff \rho(p) \leq \rho(q).$$
But the OP is interest in strict relations, that is
\begin{equation}
p< q \iff \rho(p) < \rho(q).\label{equation}\tag{$\dagger$}
\end{equation}
This makes a weaker statement, and $\rho$ no longer needs to be injective, and so not an order-embedding.
An antichain $A$ is a poset where, for $a,b \in A$ we have
$$a\leq b \iff a=b.$$
Let $P$ be a poset of length $n$ (that is, the longest chain in $P$ has $n+1$ elements).
Then $P$ satisfies \eqref{equation} iff there exist antichains $A_0,A_1, \ldots, A_n$ such that
$$P=A_0 \oplus A_1 \oplus \cdots \oplus A_n,$$
where the operation $X\oplus Y$, for posets $(X,\leq_X)$ and $(Y,\leq_Y)$, is defined by
$$
u\leq v \iff
\begin{cases}
u,v\in X \text{ and } u \leq_X v, \text{ or}\\
u,v\in Y \text{ and } u \leq_Y v, \text{ or}\\
u\in X \text{ and } v \in Y.
\end{cases}
$$
