Let $X$ be a non-empty set and let $\preceq$ be a preorder on $X$.
Is there a standard name for the induced equivalence relation '$\equiv$' on $X$ that satisfies: for every $x, y \in X$, $x \equiv y$ iff $x \preceq y$ and $x \succeq y$? Is there a standard name for the collection of equivalence classes associated with '$\equiv$'?
Let $Y$ be a non-empty set. Is there a standard name for those functions $f:X\rightarrow Y$ that respect the induced equivalence relation described above, i.e. those functions satisfying: whenever $x, y \in X$ are such that $x \preceq y$ and $x \succeq y$, $f(x) = f(y)$?