Proposition 2

Tag 03P7, permanent. Cite it as [Bourbaki, Tag 03P7].

Theory of Sets, chapter I, Appendix 0. Characterization of terms and relations

A word is significant if and only if it is balanced.

Let $A$ be a significant word belonging to a significant sequence

A_1, \; A_2, \; \ldots, \; A_n.

We shall show by induction on $k$ that each $A_k$ is balanced. Suppose that this has been established for the $A_j$ with index $j < k$ , and let us show that it is true for $A_k$ . If $A_k$ is a sign of weight 0 (which is the only possibility when $k = 1$ ), $A_k$ is balanced because

l(A_k) = 1 \qquad \text{and} \qquad n(A_k) = 0.

If $A_k$ is not a sign of weight 0, then $A_k = fB_1B_2 \ldots B_p$ , where $f$ is a sign of weight $p$ , and the $B_j$ are of the form $A_{i_j}$ , where $i_j < k$ , and are therefore balanced words by the inductive hypothesis. We have

\begin{aligned} l(A_k) &= 1 + l(B_1) + l(B_2) + \cdots + l(B_p) \\ &= 1 + (n(B_1) + 1) + (n(B_2) + 1) + \cdots + (n(B_p) + 1) \\ &= 1 + p + n(B_1) + n(B_2) + \cdots + n(B_p) \\ &= 1 + n(A_k). \end{aligned}

Let $C$ be a proper initial segment of $A$ , and let $q$ be the largest of the integers $m < p$ such that $B_m$ is a segment of $C$ , so that

C = f B_1 B_2 \ldots B_q D,

where $D$ is a proper initial segment of $B_{q+1}$ . Then

\begin{aligned} l(C) &= 1 + l(B_1) + l(B_2) + \cdots + l(B_q) + l(D) \\ &\leqslant 1 + (n(B_1) + 1) + (n(B_2) + 1) + \cdots + (n(B_q) + 1) + n(D) \\ &\leqslant p + n(B_1) + \cdots + n(B_q) + n(D) = n(C). \end{aligned}

Hence $A_k$ is balanced.

¶ To prove that, conversely, every balanced word is significant, we need the following two lemmas :

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Appendix 0. Characterization of terms and relations

Cites

Nothing.