What is the name for this process of creating an equivalence relation from a given relation? I have a relation $\sim$ on $X$ that is reflexive and symmetric. I want to form an equivalence relation $\approx$ from $\sim.$ For my purposes, the transitive closure of $\sim$ is too coarse, that is, its classes are too large. So I define $x\approx y$ if $x\sim y$ and for all $z\in X,$ we have $x\sim z$ if and only if $y\sim z.$ The relation $\approx$ so defined is an equivalence relation that is finer than the transitive closure of $\sim.$
Is there a name for the process I described above? Basically you throw out all relations that do not respect transitivity, rather than throw in extra relations to assure transitivity, the latter process being "taking the transitive closure".
I also suspect that $\approx$ is the coarsest equivalence relation that is finer than $\sim,$ but I'm not sure that that is true and it is not important for my purposes.
 A: To address your final paragraph first, no, your construction does not necessarily give "the coarsest equivalence relation contained in $\sim$" among other reasons because there need not be such a thing.
The smallest counterexample is the following: take $X=\{a,b,c\}$, and let $\sim$ be the relation
$$\sim = \{(a,a), (b,b), (c,c), (a,b), (b,a), (b,c), (c,b)\}.$$
Your construction gives $\approx = \{(a,a), (b,b), (c,c)\}$ (which is easy to see with the description of $\approx$ I give below). However, there are two incomparable equivalence relations that properly contain $\approx$ and are contained in $\sim$, namely:
$$\{(a,a),(a,b),(b,a),(b,b),(c,c)\}\quad\text{and}\quad\{(a,a),(b,b), (b,c), (c,b), (c,c)\}.$$
Now, given a relation $R$ on $X$, we can define for each $x\in X$ the sets
$$\begin{align*}
x_R &= \{z\in X\mid (x,z)\in R\},\\
{}_Rx &= \{z\in X\mid (z,x)\in R\}.
\end{align*}$$
Then we can define the equivalence relations $\approx_R$ and ${}_R\approx$ by
$$\begin{align*}
x\approx_R y&\iff x_R=y_R;\\
x{}_R\!\approx y &\iff {}_Rx={}_Ry.
\end{align*}$$
If $R$ is symmetric, then clearly $x_R={}_Rx$, and $\approx_R={}_R\!\approx$. In this case we can simply denote the resulting equivalence relation by $\approx$, if $R$ is understood from context.
If $R$ is reflexive, then we have $\approx_R\subseteq R$ and ${}_R\approx\subset R$: indeed, if $x\approx_R y$, then since $(y,y)\in R$, it follows that $y\in y_R=x_R$, so $(x,y)\in R$. Symmetrically, if $x{}_R\!\approx y$, then since $x\in {}_Rx={}_Ry$, it follows that $(x,y)\in R$.
Your construction is $\approx$ as described above. Since you are assuming $\sim$ is reflexive, you can simplify the definition by saying
$$x\approx y\iff \forall z\in X (x\sim z\leftrightarrow y\sim z)$$
since the fact that $x\sim y$ will follow from $x\approx y$.
I don't know if there is a name for this construction, but here is a way to visualize it: think of $\sim$ as the adjacency relation on a graph with vertex set $X$. Let $\Gamma$ be the resulting graph. Now perform an "edge contraction" continuous deformation, where you shrink the edge $[x,y]$ and make the vertices $x$ and $y$ coincide if the set of immediate neighbors of $x$ is the same as the set of immediate neighbors of $y$ (that is, the vertices $x$ and $y$ are adjacent to the same vertices). If we let $\Gamma/\sim$ be the resulting graph, then your equivalence relation is the "kernel" of the quotient map $\phi\colon\Gamma\to \Gamma/\sim$, where as in semigroup theory the kernel $\ker\phi$ is the congruence relation $\ker\phi\subseteq X\times X$ given by
$$\ker\phi =\{ (x,y)\in X\times X\mid \phi(x)=\phi(y)\}.$$
I'm pretty sure this equivalence relation is known to graph theorists/topologists, but I don't know what name they give it.
