TITLE: Generalizing The Cardinal Characteristics of the Continuum
QUESTION [5 upvotes]: 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 $\mathfrak{d}_\omega$.

REPLY [3 votes]: 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.