An inverse of the transitive closure

Question

Let $R\subseteq X\times X$ denote a binary relation on the set $X$. Let $\mathrm{trans}(R)$ denote the transitive closure of $R$, that is, the inclusion-wise minimal transitive relation that contains $R$.

Now, consider the following definition, which is intended to be some kind of inverse of $\mathrm{trans}(\cdot)$.

Definition. If $R$ is transitive then let $R^*\subseteq R$ be an inclusion-wise minimal relation with $\mathrm{trans}(R^*)=R$.

I am aware, that $R^*$ might not exist, e.g. for $(\Bbb Q,<)$. Here is an example, where it exists: if $R$ is the natural order relation on $\Bbb Z$, then $R^*=\{(n,n+1)\mid n\in\Bbb Z\}$.

I have two questions:

1. If $R$ is a partial order and $R^*$ exists, is it unique?
2. Has this construction a name and can be found in the literature?

Answer 1

You are looking for transitive reductions. If $R$ is a partial order with finite intervals, you get exactly the covering relations that uniquely define the partial order via its Hasse diagram.

Answer 2

I don't think that the $R^*$ will be unique. Take as an example a graph on the vertex set $\{1,2,3,4\}$ and let $R$ be the set of all possible edges on this set. Any path of three edges that is not a cycle is a candidate for $R^*$ if I understand correctly.