Say I want to tease out an arithmetical property from the ether: I get it's $\Pi^0_n$ $(\Sigma^0_n)$, and I get some indices for programs that compute structures (their (atomic) diagrams) with that property. So lets say I want to check for a certain property among those guys, and I manage to show this is $\Pi^0_m$ $(\Sigma^0_m)$ relative to this set of indices I was talking about.
Is this new property $\Pi^0_{m + n}$ $(\Sigma^0_{m+n})$ in some absolute sense? If there is already some terminology for this (probably yes) please let me know. Also if necessary please smuggle in the word complete where suitable.
Yes, everything is basically additive. However, there's a slight catch:
Suppose $A$ is $\Sigma^0_1$ and $B$ has the form $$\{x: \exists y\varphi(x,y)\},$$ where $\varphi$ is a formula with only bounded quantifiers in which $A$ gets used as a unary predicate symbol. Now there are two plausible guesses for the complexity of $B$, namely $\Sigma^0_1$ (since everything is existential) or $\Sigma^0_2$ (since there are two quantifiers involved).
The answer depends on how the unary predicate corresponding to $A$ gets used in $\varphi$. If it occurs only positively, then $B$ is $\Sigma^0_1$; otherwise, however, $B$ is merely guaranteed to be $\Sigma^0_2$. Indeed you can show that the following are equivalent for a set $X\subseteq\mathbb{N}$:
$X$ is $\Sigma^0_2$.
$X$ is c.e. relative to some co-c.e. oracle.
$X$ is c.e. relative to some c.e. oracle.
The point is that in a Turing reduction, both positive and negative information about the oracle is used (contrast this with enumeration reductions).
This indicates a dangerous ambiguity with phrases like "$\Sigma^0_m$-over-$\Sigma^0_k$" - we need to be careful about whether the inner predicate occurs positively or both positively and negatively. From what you've described, the latter is the correct interpretation. We then have pretty much what one should expect, e.g. in this sense $\Sigma^0_m$-over-$\Sigma^0_n$ is $\Sigma^0_m$-over-$\Pi^0_n$ is $\Sigma^0_{m+n}$. But in other contexts we might want to pay attention to finer restrictions and have "positivity" conditions involved in the way the inner predicate gets used.
However, basically everything else gets incredibly messy. For example:
Suppose $X$ is a high c.e. set, $X\le_m Y$, and $Y$ is "$\Pi^0_1$-complete-over-$X$." Then $X$ is not $\Sigma^0_1$ complete but $Y$ is $\Pi^0_2$-complete.
That "$X\le_mY$" bit is important: we can have e.g. a c.e. set $X$ and a set $Y$ such that $Y$ is "c.e.-over-$X$" but $Y$ doesn't compute $X$!
So folding in e.g. completeness questions makes things very fiddly.