Is there a standard name for a (multiplicatively-written) semigroup $(A, \cdot)$ such that, given an arbitrary $a \in A$, the equation $x^n = a$ has at least one solution $x \in A$? And what about the case where such a solution is required to be unique? Unluckily I can find no trace of them in the two volumes of Clifford and Preston's survey: The algebraic theory of semigroups, but this is almost surely for the fact that I don't know what to look for. Thus, I would appreciate much if you could also provide me with references to previous work on the subject.

Is there a standard name for a (multiplicatively-written) semigroup $(A, \cdot)$ such that, given an arbitrary $a \in A$, the equation $x^n = a$ has at least one solution $x \in A$ for each $n \in \mathbb{N}^+$? And what about the case where such a solution is required to be unique? Unluckily I can find no trace of them in the two volumes of Clifford and Preston's survey: The algebraic theory of semigroups, but this is almost surely for the fact that I don't know what to look for. Thus, I would appreciate much if you could also provide me with references to previous work on the subject.

P.S. If I'm not blinded by the dragon's breath, nothing similar seems to be there in Wiki's article on Special classes of semigroups (click).