I have an operad with object sets $P(n)$. For most $n$, there exists at least one object $x\in P(n)$ such that for all sets of other objects and all compositions thereof, $x$ is not equal to that composition.
Is there a convention for what to call $x$? An atom? An atomic object?
I believe $x$ would typically be called "indecomposable", like in algebras.
Note that in order to define correctly what indecomposable means, you need to realize that you can always write $x = \operatorname{id}(x) = x(\operatorname{id},\dots,\operatorname{id})$ where $1$ is the identity of your operad. So you need to discard such trivial compositions.
If you have an operad of sets this works fine, in more general settings you probably need to consider an augmented operad (i.e. an operad equipped with a map $\varepsilon : P(1) \to I$ where $I$ is the unit of your monoidal category, satisfying $\varepsilon \eta = \operatorname{id}_I$ where $\eta : I \to P(1)$ represents the unit). Then an indecomposable is something that isn't in the image of $\bar{P} \circ_{(1)} \bar{P} \to P$, where $\bar{P}$ is the augmentation ideal.