Remark

Let $S$ be a subset of $X$. Recall that we have identified $M(S)$ with a submagma of $M(X)$, $Mo(S)$ with a submonoid of $Mo(X)$ and $N^{(S)}$ with a submonoid of $N^{(X)}$. Then

Clearly $\lambda(M(S)) \subset Mo(S)$. Let $w \in \lambda^{-1}(Mo(S))$; we show by induction on $l(w)$ that $w \in M(S)$. It is obvious if $l(w) = 1$. If $l(w) > 1$, we may write $w = w_1 w_2$ with $w_1, w_2 \in M(X)$, $l(w_1) < l(w)$, $l(w_2) < l(w)$. Then $\lambda(w_1) \lambda(w_2) \in Mo(S)$, hence $\lambda(w_1) \in Mo(S)$ and $\lambda(w_2) \in Mo(S)$, whence $w_1 \in M(S)$ and $w_2 \in M(S)$ by the induction hypothesis and finally $w \in M(S)$.

Also

This follows immediately from formula (21).

Further, $N^{(S)}$ is the set of elements of $N^{(X)}$ whose support is contained in $S$; if $(S_i)_{i \in I}$ is a family of subsets of $X$ of intersection $S$, then $N^{(S)} = \bigcap_{i \in I} N^{(S_i)}$ and formulae (23) and (24) imply