Generators of families of langauges?
For the sake of notation, let's call a family of languages $\mathcal{L}$ that contains


the empty set $\emptyset$ and
all singleton sets $\{a\}$, $a$ some symbol,


a founded family, and 

$\qquad\displaystyle \mathcal{L}_+ = \{ L \in \mathcal{L} \mid \varepsilon \not\in L\}$.

Let $L \in \mathrm{REG}_+$ arbitrary. We will construct $L$ from only the empty and singleton sets using the trio operations


$\varepsilon$-free homomorphism,
inverse homomorphism and
intersection with regular languages,


thus proving that (1) all founded trios also contain $\mathrm{REG}_+$. Furthermore, (2) $\mathrm{REG}_+$ it is a founded trio itself. Therefore, we have shown that $\mathrm{REG}_+$ is the minimal founded trio.




It suffices to show that $\Sigma^+ \supset L$ is contained in every founded trio $\mathcal{T}$; the rest is trivial by $L = \Sigma^+ \cap L \in \mathrm{REG}_+$.

We define for any alphabet $\Sigma$ a marked copy $\hat{\Sigma} = \{ \hat{a} \mid a \in \Sigma \}$ and two homomorphisms $h,g : \Sigma \cup \hat{\Sigma} \to \Sigma^*$ by

$\qquad\displaystyle 
  h(a) = \begin{cases}
           a &amp;, a \in \Sigma \\
           \varepsilon &amp;, a \in \hat{\Sigma}
         \end{cases}
$

and 

$\qquad\displaystyle 
  g(a) = \begin{cases}
           b &amp;, a = b \in \Sigma \\
           b &amp;, a = \hat{b} \in \hat{\Sigma}
         \end{cases}
$.

Now consider

$\qquad\displaystyle L' = \Sigma \hat{\Sigma}^*$.

By noting that


$\Sigma = h(L') \iff h^{-1}(\Sigma) = L'$,
$g(L') = \Sigma^+$ and
$g$ is $\varepsilon$-free,


we obtain from the trio closure properties that

$\qquad\displaystyle \Sigma \in \mathcal{T} \implies \Sigma^+ \in \mathcal{T}$.

But since $\Sigma$ is the pre-image of every $\{a\} \subseteq \Sigma$ with suitable constant homomorphism $h_a(b) = a$ we have $\Sigma \in \mathcal{T}$ by foundedness and closure under inverse homomorphism.
We check the conditions of a founded trio for $\mathrm{REG}_+$:


Closure under $\varepsilon$-free homomorphism: clear since $\mathrm{REG}$ has this property and no such homomorphism can generate $\varepsilon$ from an $\varepsilon$-free language.
Closure under inverse homomorphism: clear since $\mathrm{REG}$ has this property and the image of $\varepsilon$ is always $\varepsilon$.
Closure against intersection with regular languages: clear by def.
Foundedness: clear by def; all required sets are regular and don't contain the empty word..





$\mathrm{REG}$ actually is the minimal "founded" trio if you add $\{\varepsilon\}$ to the base; a similar proof goes through.

Furthermore, $\mathrm{REG}_+$ is not a full trio (cone) because of it's lack of empty words. We extend above proof by noting that containment of $\mathrm{REG}_+$ now implies containment of $\mathrm{REG}$ (any regular language $L$ is the image of $\hat{a}L \in \mathrm{REG}_+$ by a homomorphism that deletes only $\hat{a}$). Since $\mathrm{REG}$ is itself a cone it is also the minimal cone.



The properties extend to (semi-)AFLs, i.e. $\mathrm{REG}_+$ is the minimal (semi-)AFL and $\mathrm{REG}$ is the minimal full (semi-)AFL.