Processes and structures on approximation spaces

We introduce the concept of a computability component on an admissible set and consider minimal and maximal computability components on hereditarily finite superstructures as well as jumps corresponding to these components. It is shown that the field of real numbers $\Sigma$-reduces to jumps of the maximal computability component on the least admissible set $\mathbb{HF}(\varnothing)$. Thus we obtain a result that, in terms of $\Sigma$-reducibility, connects real numbers, conceived of as a structure, with real numbers, conceived of as an approximation space. Also we formulate a series of natural open questions.