Has any work been done on generalizing any characteristics of the cardinality of the continuum?
The bounding number $\mathfrak{b}$ and dominating number $\mathfrak{d}$ could be easily generalized for ordinals $\alpha$, as follows:
$$\forall f,g\in\alpha^\alpha(f\leq_{\alpha}g\Leftrightarrow|\{\beta\in\alpha:f(\beta)>g(\beta)\}|<|\alpha|)$$ $$\mathfrak{b}_\alpha=\min\{|F|:F\subseteq\alpha^\alpha\land\forall f\in\alpha^\alpha\exists g\in F(g\not\leq_\alpha f)\}$$ $$\mathfrak{d}_\alpha=\min\{|F|:F\subseteq\alpha^\alpha\land\forall f\in\alpha^\alpha\exists g\in F(f\leq_\alpha g)\}$$
A few points to be made at this point:
$\mathfrak{b}_\alpha\leq\max\{|F|:F\subseteq\alpha^{\alpha}\}=|\alpha^\alpha|=2^{|\alpha|}$
For cardinals $\kappa,\mathfrak{b}_\kappa>\kappa$ (this is shown by generalized diagonalization)
The last two points combined make $\mathfrak{b}_\kappa=\kappa^+$ if $\kappa^+=2^\kappa$ (or if GCH is assumed)
$\mathfrak{b}_\omega$ is clearly, in this case, $\mathfrak{b}$, and the same is true with $\mathfrak{d}_\omega$.
