Absolutely summable signals
Absolute summability implies that the Fourier transform converges uniformly, so this means that the unit circle is contained in the region of convergence (ROC).

Now, the pole at 0.5 is the limit of the ROC, so the ROC is from 0.5 outwards (in order to include the unit circle). This means that the signal is right-sided.

Finite support signals cannot have poles (other than $z=0$ or $z=\infty$).