Let $S$ be the set of non-negative rational numbers. (If it makes any difference, feel free to take the non-negative dyadic rationals instead.) Let $B=\ell_\infty(S)$; as a ${\rm C}^*$-algebra this is isomorphic to $\ell_\infty({\bf N})$ and so the Gelfand spectrum of $B$ is homeomorphic to $\beta{\bf N}$.

Now let $$A=\{ f \in B \colon f(t) \to 0 \hbox{ as $t\to+\infty$} \}.$$ (That is, for all $\varepsilon>0$ there exists $K\in [0,\infty)$ such that $|f(t)| < \varepsilon$ whenever $t\in S\cap (K,\infty)$.) This is a closed ideal in $B$, in particular it's an abelian ${\rm C}^*$-algebra so we can consider its Gelfand spectrum $\Phi_A$.

Let $\Omega_S$ be the one-point compactification of $\Phi_A$, which is homeomorphic to a quotient of $\beta {\bf N}$.

Question.Does this compactification of thediscretespace $S$ have a name in the literature, and does it admit any description more concrete than saying "it consists of equivalence classes of certain ultrafilters"?

I guess that one could make a more abstract version of the question by looking at a total order on a countable discrete space $T$ and considering the corresponding compactification of $T$. Note that I'm not considering $T$ with the order topology, but the discrete topology.