Exercise 2, Appendix 0

In a linearly ordered set $I$, we shall say that a non-empty subset $J$ is a slice of $I$ if, whenever $\alpha$ and $\beta$ are elements of $J$ such that $\alpha < \beta$, then $[\alpha, \beta] \subset J$.

a) Let $(x_i)_{i \in J}$ be a multipliable family in a complete group $G$. Show that, for every slice $J$ of $I$, the family $(x_i)_{i \in J}$ is multipliable (consider first the particular case in which the set of lower bounds of $J$ which do not belong to $J$ is empty).

b) A partition $(J_\lambda)_{\lambda \in L}$ of $I$ is said to be an ordered partition if $L$ is a linearly ordered set and if the relations $\lambda < \mu, \alpha \in J_\lambda, \beta \in J_\mu$ imply $\alpha < \beta$; the $J_\lambda$ are then slices of $I$. If $(x_i)_{i \in I}$ is a multipliable family in a complete group $G$, show that if we put $p_\lambda = \prod_{i \in J_\lambda} x_i$, then the family $(P_\lambda)_{\lambda \in L}$ is multipliable and has the same product as the family $(x_i)_{i \in I}$.

c) Conversely, if $(J_\lambda)_{\lambda \in L}$ is a finite ordered partition of $I$, and if $(x_i)_{i \in I}$ is such that each of the families $(x_i)_{i \in J_\lambda}$ is multipliable, then the family $(x_i)_{i \in I}$ is multipliable.