Recursive, non-recursive systems; FIR, IIR systems
The logical implications are the following:
&quot;non-recursive&quot; $\Longrightarrow$ FIR
IIR $\Longrightarrow$ &quot;recursive&quot;
But the opposites are not necessarily true because a FIR system can be implemented recursively (transfer function poles can be cancelled by zeros).

Of course, when referring to "recursive" or "non-recursive" we always talk about implementations with finitely many operations per output sample. Clearly, any discrete-time LTI system can be described by a generally infinite convolution sum, but that is not what we mean by "non-recursive".