# z-transform causality properties: negative coefficents are zero ($x[-1]z^1=0$, $x[-2]z^2=0$, …)

Let's suppose I have a system:

$$Y(z)=X(z)H(z)$$

If the system is causal, does that mean that all the negative coefficients (example: x[-1]) of the transform for $$Y(z)$$, $$X(z)$$, and $$H(z)$$ are zero? or does that rule apply only to $$H(z)$$? what about $$Y(z)$$ and $$X(z)$$?

No. The system response is usually $$H(z)$$. If $$H$$ is causal, then it has no non-causal coefficients. $$X$$ may very well be a signal that exists for all time (and so existed before time 0) and, therefore, have non-causal coefficients. That means $$Y$$ will also have non-causal coefficients.
If $$Y$$ is taken as the system and $$Y$$ is causal, then your statement is true. But usually $$Y$$ is the output, $$X$$ is the input, and $$H$$ is the system response.