“topological fellows” is often indirect, though sometimes very surprising. For example, if we define $${{\rm Depth}}_{{\rm H}+}({{\mathbb B}})=\sup\{{{\rm Depth}}({{\mathbb B}}/I): I\mbox{ is an ideal in }{{\mathbb B}}\;\},$$ then for any (infinite) Boolean algebra ${{\mathbb B}}$ we will have that ${{\rm Depth}}_{{\rm
H}+}({{\mathbb B}})$ is the tightness $t({{\mathbb B}})$ of the algebra ${{\mathbb B}}$ (or the tightness of the topological space ${\rm Ult}({{\mathbb B}})$ of ultrafilters on ${{\mathbb B}}$), see [@M2 Theorem 4.21]. A somewhat similar function to ${{\rm Depth}}_{{\rm H}+}$ is obtained by taking $\sup\{{{\rm Depth}}({{\mathbb B}}'): {{\mathbb B}}'$ is a subalgebra of ${{\mathbb B}}\;\}$, but clearly this brings nothing new: it is the old Depth. But if one wants to understand the behaviour of the depth for subalgebras of the considered Boolean algebra, then looking at the following [*subalgebra ${{\rm Depth}}$ relation*]{} may be very appropriate: $$\begin{array}{lr}
{{\rm Depth}}_{\rm Sr}({{\mathbb B}})=\{(\kappa,\mu):&\mbox{there is an infinite subalgebra
${{\mathbb B}}'$ of ${{\mathbb B}}$ such that }\ \\
&|{{\mathbb B}}'|=\mu\mbox{ and }{{\rm Depth}}({{\mathbb B}}')=\kappa\;\}.
  \end{array}$$ A number of results related to this relation is presented by Monk in [@M2 Chapter 4]. There he asks if there are a Boolean algebra ${{\mathbb B}}$ and an infinite cardinal $\theta$ such that $(\theta,(2^\theta)^+)\in {{\rm Depth}}_{\rm
Sr}({{\mathbb 