and }\quad
    (\forall g\in F^*)(g{\restriction}w\in{{\rm cl}}(F))$$ then ${{\mathbb B}}_{(w,F)}$ is a subalgebra of ${{\mathbb B}}_{(w^*,F^*)}$.

So each condition $p$ in our forcing notion ${{{\mathbb P}^\theta_\lambda}}$ will have a set $u^p\in
[\lambda^+]^{<\lambda}$ and a closed set $F^p\subseteq 2^{\textstyle u^p}$ (and the respective algebra will be ${{\mathbb B}}_p={{\mathbb B}}_{(u^p,F^p)}$). But to make the forcing notion work, we will have to put more restrictions on our conditions, and we will be taking only those conditions that have to be taken to make the arguments work. For example, we want that cardinals are not collapsed by our forcing, and demanding that ${{{\mathbb P}^\theta_\lambda}}$ is $\lambda^+$-cc (and somewhat $({<}\lambda)$–closed) is natural in this context. How do we argue that a forcing notion is $\lambda^+$–cc? Typically we start with a sequence of $\lambda^+$ distinct conditions, we carry out some “cleaning procedure” (usually involving the $\Delta$–lemma etc), and we end up with (at least two) conditions that “can be put together”. Putting together two (or more) conditions that are approximations to a Boolean algebra means amalgamating them. There are various ways to amalgamate conditions - we will pick one that will work for several purposes. Then, once we declare that some conditions forming a “clean” $\Delta$–sequence of 