On the definition of the reflexive-transitive hull 
The reflexive-transitive closure of a binary relation  is the smallest reflexive and transitive binary relation containing .

Sentences like the above one are simultaneously widespread and meaningless, since a standalone “reflexive” is not defined.  What is defined instead is “reflexive on a set ”.  Having said that, we are left with a few possibilities to convert the above definition into a proper one:
(A) Given a binary relation  on a set , the reflexive-transitive closure of  with respect to  is the smallest binary relation containing  that is reflexive on  and transitive.
(B) The reflexive-transitive closure of a binary relation  is the smallest binary relation containing  that is reflexive on π₁() and transitive.
(C) The reflexive-transitive closure of a binary relation  is the smallest binary relation containing  that is reflexive on π₂() and transitive.
(D) The reflexive-transitive closure of a binary relation  is the smallest binary relation containing  that is reflexive on π₁()∪π₂() and transitive.  Equivalently, the reflexive-transitive closure of a binary relation  is the smallest preorder on π₁()∪π₂() containing .
Here, πᵢ() is the projection of  on the th component (∈{1,2}).
Note that (B), (C), and (D) work when a binary relation is simply defined as a set of pairs (sc., without specifying the underlying domain).
I often see (A) explicitly or implicitly (in various wordings).  Does any scholar work or a mechanized theory use (B), (C), or (D)?
 A: It's clearly that (D) is the right interpretation on the sentence in the block. In fact, given any binary relation $R$ we can produce its reflexive-transitive closure in the following way:
\begin{align*}
R_0&=\textstyle R\cup\{(x,x)\mid x\in \mathrm{field}(R)\},\\
R_1&=\textstyle R_0\cup\{(x,z)\mid x,z\in \mathrm{field}(R)\wedge\exists y\in \mathrm{field}(R)((x,y)\in R_0\wedge(y,z)\in R_0))\},\\
\vdots~~&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\vdots\\
R_{n+1}&=\textstyle R_n\cup\{(x,z)\mid x,z\in \mathrm{field}(R)\wedge\exists y\in \mathrm{field}(R)((x,y)\in R_n\wedge(y,z)\in R_n))\},\\
\overline{R}&=\textstyle\bigcup_{n\in\mathbb{N}}R_n,
\end{align*}
where $\mathrm{field}(R)=\bigcup(\bigcup R)=\pi_1(R)\cup\pi_2(R)$ in fact. It's easy to see that $\overline{R}$ (1) contains $R$, and (2) is reflexive and transitive, and is the smallest satisfying (1) and (2).
And if (A) is the meaning what the author intended to express, then the construction could be rivised as follows
\begin{align*}
R_0&=\textstyle R\cup\{(x,x)\mid x\in X\},\\
R_1&=\textstyle R_0\cup\{(x,z)\mid x,z\in X\wedge\exists y\in X((x,y)\in R_0\wedge(y,z)\in R_0))\},\\
\vdots~~&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\vdots\\
R_{n+1}&=\textstyle R_n\cup\{(x,z)\mid x,z\in X\wedge\exists y\in X((x,y)\in R_n\wedge(y,z)\in R_n))\},\\
R^*&=\textstyle\bigcup_{n\in\mathbb{N}}R_n.
\end{align*}
Then $R^*$ is the smallest binary relation containing $R$ which is reflexive and transitive on $X$.
