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 $\mathfrak{d}\_\omega$.



Generalizing The Cardinal Characteristics of the Continuum
Don Monk has a paper describing generalized $\mathfrak b$ and $\mathfrak d$ as you describe. In his paper he further generalizes to $\mathfrak b\_{\kappa,\lambda,\mu}$ (and analogously for $\mathfrak d$), considering families of functions in ${}^\lambda\mu$ with $\lambda, \mu,$ and $\kappa$ all possibly distinct. I think in his notation the numbers you describe are $\mathfrak b\_{\alpha, \alpha, \alpha}$ and $\mathfrak d\_{\alpha, \alpha, \alpha}$


[Generalized ${\frak b}$ and ${\frak d}$. Notre Dame J. Formal Logic 45 (2004), no. 3, 129--146.](http://euclid.colorado.edu/~monkd/monk75.pdf) 


