How to call a partition of $X$ which consists of all singleton subsets of $X$?

Question

In other words, if $X$ is a set, then how do we call $Y=\{\{x\}:x\in X\}$?

$\{X\}$ is already named the trivial partition, so that cannot be it. Complete partition and total partition did not yield enough search results. Does it maybe not have a name?

A related question: does $Y$ exist for every $X$?