Characterisation of a poset 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 $aRb\Leftrightarrow f(x)Qf(y)$.)
 A: 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 $1<5,6$; $2<5,6$; $3<7,8$; $4<7,8$.
A: First you would want the multiset and not the set. Even then the answer is no. I'm sure this would follow from counting considerations ( For $n$ large enough there should be wildly many more posts or even lattices than there are possible lists of pairs.) But simple examples exist.
If the set was $\{{(1,2),(2,1)\}}$ that would say that we essentially have a graph consisting of one or more disjoint cycles and the poset consisting of the points and edges. Then the number of points and edges would be equal, but unknown. 
If we had the multiset with each of $(1,2)$ and $(2,1)$ nine times then we would know that the graph was either a $9$-gon OR a square and a pentagon OR three triangles OR a triangle and a hexagon.  But we would not know which. 
To make this into a lattice, use $(1,19)$ and $(19,1)$ once each along with $(2,3)$ and $(3,2)$ nine times each.
I'm sure that even knowing one had a lattice $L$ with $k$ levels and the multiset of all sublattices with $k-1$ levels would not be enough to describe $L$ up to isomorphism. The little example above is one counter-example for $k=4.$
