The reason why the minimal dense sublocale of a frame is important is because it is a complete Boolean algebra and complete Boolean algebras are very special kinds of frames. Furthermore, the nucleus corresponding to a minimal dense sublocale gives an example of a frame homomorphism $f:L\rightarrow B$ and we shall see why such homomorphisms are essential to point-free topology. In this post, I will mainly talk about the importance of frame homomorphisms $f:L\rightarrow B$ for the sake of generality.

Complete Boolean algebras make point-free topology much more elegant than it would otherwise be, and the minimal dense sublocale of a frame is an important part of the relation between complete Boolean algebras and point-free topology.

Complete Boolean algebras satisfy some of the highest separation axioms, and they even satisfy some other notable peculiar point-free topological properties. Complete Boolean algebras are always regular, completely regular, normal, zero-dimensional, paracompact, ultraparacompact, extremally disconnected, $P$-frames, etc. Complete Boolean algebras are also notable because all atomless complete Boolean algebras are completely point-free (the points in a complete Boolean algebra are precisely the generic ultrafilters on that complete Boolean algebra).

If $\kappa$ is an uncountable regular cardinal, then we say a regular space $(X,\mathcal{T})$ is a $P_{\kappa}$-space if the intersection of less than $\kappa$ many open sets is still open. We say that a frame $L$ is weakly $\kappa$-distributive if it satisfies the property $x\vee\bigwedge_{i\in I}y_{i}=\bigwedge_{i\in I}(x\vee y_{i})$ whenever $|I|<\kappa$. Therefore, the notion of a weakly $\kappa$-distributive frame is a generalization of the notion of a $P_{\kappa}$-space (Caution: We need to be careful since there are multiple inequivalent but natural ways of generalizing the notion of a $P_{\kappa}$-space to point-free topology).

$\textbf{Theorem:}$ A regular space $(X,\mathcal{T})$ is a $P_{\kappa}$-space if and only if the frame $\mathcal{T}$ is weakly $\kappa$-distributive.

$\textbf{Theorem:}$ A regular frame $L$ is a complete Boolean algebra if and only if it is weakly $\kappa$-distributive for all uncountable regular cardinals $\kappa$.

One should think of a complete Boolean algebra as therefore like a space where the arbitrary intersection of open sets is open, and complete Boolean algebras in a sense behave like discrete spaces. In fact, every sublocale of a complete Boolean algebra is both an open and a closed sublocale. A regular frame is a complete Boolean algebra precisely when it has no proper dense sublocale.

If $B_{L}$ is the minimal dense sublocale of a frame $L$, then there is a surjective frame homomorphism $L\rightarrow B_{L}$ defined by $x\mapsto x^{**}$.

Let $\mathfrak{b}(a)$ denote the smallest dense sublocale of $L$ containing $a$. Then there is a surjective frame homomorphism
$\phi_{a}:L\rightarrow\mathfrak{b}(a)$ defined by $\phi_{a}(x)=(x\rightarrow a)\rightarrow a.$ Then $\mathfrak{b}(a)$ is a complete Boolean algebra and if $S\subseteq L$ is a sublocale which is a complete Boolean algebra, then $S=\mathfrak{b}(a)$ for some $a$.

Fact: Every frame $L$ embeds as a subframe of a product of complete Boolean algebras. In particular, the mapping $\phi:L\rightarrow\prod_{a\in L}\mathfrak{b}(a)$ defined by $\phi(x)=(\phi_{a}(x))_{a\in L}$ is an embedding.

The above fact in a sense of the extension of the fact that every topology $(X,\mathcal{T})$ embeds as a subframe of $\{0,1\}^{X}$. One could also show that every regular frame $L$ is a subframe of a product of complete Boolean algebras using forcing, but such an argument is more difficult and less efficient in terms of the cardinality of the complete Boolean algebras required.

**The congruence tower**

We shall now describe a construction that allows us to extend a frame $L$ to a much larger frame $M$ but where the frame homomorphisms $f:L\rightarrow B$ are in a one-to-one correspondence with the frame homomorphisms $g:M\rightarrow B$. This construction is only interesting because for each frame $L$ there are many interesting frame homomorphisms $f:L\rightarrow B$ including the frame homomorphism from $L$ onto the minimal dense sublocale of $L.$

Suppose that $L$ is a frame. Then a congruence on $L$ is an equivalence relation $\simeq$ such that if $v\simeq w,x\simeq y$, then $v\wedge x\simeq w\simeq y$ and if $x_{i}\simeq y_{i}$ for $i\in I$, then $\bigvee_{i\in I}x_{i}\simeq\bigvee_{i\in I}y_{i}$. Let $\mathfrak{C}(L)$ denote the lattice of all congruences on the frame $L$. Then $\mathfrak{C}(L)$ is itself a zero-dimensional frame. Define a mapping $\nabla_{L}:L\rightarrow\mathfrak{C}(L)$ by letting $(x,y)\in\nabla_{L}(a)$ if and only if $x\vee a=y\vee a$. Then $\nabla_{L}$ is an injective frame homomorphism such that each $x\in L$ is complemented in $\mathfrak{C}(L)$.

$\textbf{Theorem:}$ Suppose that $L$ is a frame, $B$ is a complete
Boolean algebra, and $f:L\rightarrow B$ is a frame homomorphism. Then
there is a unique frame homomorphism $\overline{f}:\mathfrak{C}(L)\rightarrow B$ such that $f=\overline{f}\nabla_{L}$.

The above theorem would not be as interesting if frame homomorphisms $f:L\rightarrow B$ with $B$ Boolean were rare or difficult to construct.

Suppose that $L$ is a frame. Then define $\mathfrak{C}^{\alpha}(L)$ for each ordinal $\alpha$ by letting $\mathfrak{C}^{0}(L)=L$, $\mathfrak{C}^{\alpha+1}(L)=\mathfrak{C}(\mathfrak{C}^{\alpha}(L))$ and where if $\gamma$ is a limit ordinal, then $\mathfrak{C}^{\gamma}(L)$ is the direct limit of $(\mathfrak{C}^{\alpha}(L))_{\alpha<\gamma}$ where the transitional mappings are produced by the mappings of the form $\nabla_{\mathfrak{C}^{\alpha}(L)}$ and direct limits of such mappings. Let $\nabla_{L}^{\alpha}:L\rightarrow\mathfrak{C}^{\alpha}(L)$ be the canonical embedding.

$\textbf{Theorem:}$ Suppose that $L$ is a frame, $B$ is a complete
Boolean algebra, $\alpha$ is an ordinal, and $f:L\rightarrow B$ is a
frame homomorphism. Then there is a unique frame homomorphism
$\overline{f}:\mathfrak{C}^{\alpha}(L)\rightarrow B$ such that $f=\overline{f}\nabla_{L}^{\alpha}$.

We now have examples of towers of structures that, unlike the automorphism group tower, do not stop growing.

$\textbf{Theorem:}$ There is a frame $L$ such that the congruence
tower $(\mathfrak{C}^{\alpha}(L))_{\alpha}$ never stops growing.

$\textbf{Forcing and the congruence tower}$

The existence of many frame homomorphisms $\phi:L\rightarrow B$ play a very important role in how frames behave when you put them into forcing extensions and they are the basis of a field which I have worked on called Boolean-valued point free topology which is essentially about considering frames in forcing extensions (for the set theorists, since frames are complete Boolean algebras, the Boolean-valued model approach to forcing works quite well.). The frame homomorphisms $\phi:L\rightarrow B$ should be thought of as the points that you add to the frame $L$ when you pass $L$ to turn it into a frame a forcing extension using the Boolean-valued model approach to forcing.

Let $\textrm{Sp}_{B}(L)$ be the set of all frame homomorphisms $\phi:L\rightarrow B$. Then $\textrm{Sp}_{B}(L)$ is a $B$-valued structure where we set $\|\phi=\theta\|=b$ when $b$ is the largest element in $B$ with $\phi(x)\wedge b=\theta(x)\wedge b$ for each $b\in B$. Since $\textrm{Sp}_{B}(L)$ is a $B$-valued structure, one should consider $\textrm{Sp}_{B}(L)$ as an object in the Boolean-valued universe $V^{B}$.

$\textbf{Theorem}:$ Suppose that $L$ is a frame. Then the mapping
$(\nabla_{L}^{\alpha})^{*}:\textrm{Sp}_{B}(\mathfrak{C}^{\alpha}(L))\rightarrow\textrm{Sp}_{B}(L)$ is a bijection. In particular,
$$V^{B}\models\text{There is a bijective continuous function from $\textrm{Sp}_{B}(\mathfrak{C}^{\alpha}(L))$ to $\textrm{Sp}_{B}(L)$.}$$

This correspondence is startling since the frame $\mathfrak{C}^{\alpha}(L)$ could have arbitrarily large cardinality.