Let $X$ be a finite set ordered by $R$, where $R$ is a transitive, reflexive, and antisymmetric relation on $X$. We define, for all $x\in X$, $C\_R(x)=(m\_R(x),M\_R(x))\in \mathbb N^2$, such that $m\_R(x)$ is the cardinality of the set of all the minorants of $x$, and $M\_R(x)$ is the cardinality of the set of all majorants of $x$.


Does the set $\left\{C\_R(x),\, x\in X\right\}$ characterize $R$ up to isomorphism?


(We will say that $R\subset X^2$ and $Q\subset Y^2$ are isomorphic if there exist $f$, a bijection from $X$ to $Y$ such that $aRb\Leftrightarrow f(x)Qf(y)$.)



Characterisation of a poset
If I understand correctly the meaning of minorant and majorant, then the answer is negative. Let $P$ be the poset with vertices $1,\dots,8$ defined by $1<5,6$; $2<6,7$; $3<7,8$; $4<5,8$. Let $Q$ have the same vertices and be defined by $1<5,6$; $2<5,6$; $3<7,8$; $4<7,8$.


