# What does $\cdot$ mean as a notation with complexity classes?

In the wikipedia page for Toda's Theorem, the notation $A\cdot B$ is used where $A$ and $B$ are two complexity classes, but without explanation as to its meaning.

SO given two classes $A$ and $B$ what does $\cdot$ mean in $A\cdot B$?

Using an example for the cited page, what would $BP \cdot \oplus P$ mean?

Is it correct to say $A\cdot A=A$, and when is it true that $A\cdot B=B$ or $A$?

Reading Toda's original paper, it's not clear that a general application of $\cdot$ has meaning, however it may have been extended later.

Toda introduces three operators: $\oplus\cdot$, $\mathsf{BP}\cdot$ and $\mathsf{C}\cdot$, so the $\cdot$ is, in this context, not an independent piece of notation. Toda is extending the $\mathsf{BP}\cdot$ operator introduced by Schöning (but with uglier notation - $BP\mathscr{C}$ for the operator applied to class $\mathscr{C}$).

The three operators give (intuitively) the closures of classes under three different types of reductions. I'll reproduce the definition (and intuitive explanation) for $\mathsf{BP}\cdot$ for Toda's paper (p. 867):

$L \in \mathsf{BP}\cdot\mathbf{K}$ if there exist a set $A \in \mathbf{K}$, a polynomial $p$, and a constant $\alpha > 0$ such that, for all $x\in \Sigma^{\ast}$, $$\text{Prob}(\{w\in\{0,1\}^{p(|x|)}:x\#w\in A \leftrightarrow x \in L\}) \geqq \frac{1}{2}\alpha$$

The intuitive definition is (p. 865):

Intuitively speaking, a set is in $\mathrm{BP}\cdot\oplus \mathbf{P}$ if and only if it is reducible to a set in $\oplus\mathbf{P}$ under a polynomial-time randomized reduction with two-sided bounded error probability.

There is no such notation. The piece of notation you are citing is $\mathsf{BP} \cdot \mathsf{X}$, which is the bounded probabilistic version of $\mathsf{X}$, that is, $\mathsf{BP}\cdot\mathsf{X}$ has the same relation to $\mathsf{X}$ as $\mathsf{BPP}$ has to $\mathsf{P}$.