${\dot{{{\mathbb B}}}^\theta_\lambda}$, it will carry the information on what is the subalgebra ${{\mathbb B}}_p=\langle x_i: i\in u^p\rangle_{{\dot{{{\mathbb B}}}^\theta_\lambda}}$ for some $u^p\subseteq\lambda^+$. A natural way to describe algebras in this context is by listing ultrafilters (or: homomorphisms into $\{0,1\}$):

\[0.C\] For a set $w$ and a family $F\subseteq 2^{\textstyle w}$ we define

${{\rm cl}}(F)=\{g\in 2^{\textstyle w}: (\forall u\in [w]^{\textstyle
<\omega})(\exists f\in F)(f{\restriction}u=g{\restriction}u)\}$,

${{\mathbb B}}_{(w,F)}$ is the Boolean algebra generated freely by $\{x_\alpha:\alpha\in w\}$ except that

if $u_0,u_1\in [w]^{\textstyle <\omega}$ and there is no $f\in F$ such that $f{\restriction}u_0\equiv 0$, $f{\restriction}u_1\equiv 1$

then $\bigwedge\limits_{\alpha\in u_1} x_\alpha\wedge 
\bigwedge\limits_{\alpha\in u_0} (-x_\alpha)=0$.

This description of algebras is easy to handle, for example:

\[see [[@Sh:479 2.6]]{}\] \[0.D\] Let $F\subseteq 2^{\textstyle w}$. Then:

1.  Each $f\in F$ extends (uniquely) to a homomorphism from ${{\mathbb B}}_{(w,F)}$ to $\{0,1\}$ (i.e. it preserves the equalities from the definition of ${{\mathbb B}}_{(w,F)}$). If $F$ is closed, then every homomorphism from ${{\mathbb B}}_{(w,F)}$ to $\{0,1\}$ extends exactly one element of $F$.

2.  If $\tau(y_0,\ldots,y_\ell)$ is a Boolean term and $\alpha_0,\ldots,
    \alpha_\ell\in w$ are distinct then $$\begin{array}{l}
    {{\mathbb B}}_{(w,F)}\models\tau(x_{\alpha_0},\ldots,x_{\alpha_\ell})\neq 0\qquad
    \qquad\mbox{ if and only if}\\
    (\exists f\in F)(\{0,1\}\models\tau(f(\alpha_0),\ldots,f(\alpha_k))=1).
      \end{array}$$

3.  If $w\subseteq w^*$, $F^*\subseteq 2^{\textstyle w^*}$ and $$(\forall f\in F)(\exists g\in F^*)(f\subseteq g)\quad\mbox{ 