Theorem 2 (Commutativity theorem)

Tag 055Y, permanent. Cite it as [Bourbaki, Tag 055Y].

Algebra, chapter I, § 1. Laws of composition; associativity; commutativity

Let T be an associative law of composition on E; let $(x_\alpha)_{\alpha \in A}$ be a non-empty finite family of elements of E which are pairwise permutable; let B and C be two totally ordered sets with A as underlying set. Then

\prod_{\alpha \in B} x_\alpha = \prod_{\alpha \in C} x_\alpha.

Since the theorem is true if A has a single element, we argue by induction on the number $p$ of elements in A. Let $p$ be an integer > 1 and suppose the theorem is true when Card A < $p$ . We prove it for Card A = $p$ . It may be assumed that A is the interval $[0, p - 1]$ in $\mathbf{N}$ ; the composition of the ordered sequence $(x_\alpha)_{\alpha \in A}$ defined by the natural order relation on A is $\prod_{i=0}^{p-1} x_i$ .

Let A be given another total ordering and let h be the least element of A under this ordering and A' the set of other elements of A (totally ordered by the induced ordering). Suppose first $0 < h < p - 1$ and let P = $\{0, 1, \ldots, h - 1\}$ and Q = $\{h + 1, \ldots, p - 1\}$ ; the theorem being assumed true for A', applying the associativity theorem, we obtain (since $A' = P \cup Q$ )

\prod_{\alpha \in A'} x_\alpha = \left( \prod_{i=0}^{h-1} x_i \right) \circ \left( \prod_{i=h+1}^{p-1} x_i \right)

whence, composing $x_h$ with both sides and repeatedly applying the commutativity and associativity of T:

\begin{align*} \prod_{\alpha \in A} x_\alpha &= x_h \circ \left( \prod_{\alpha \in A'} x_\alpha \right) = x_h \circ \left( \prod_{i=0}^{h-1} x_i \right) \circ \left( \prod_{i=h+1}^{p-1} x_i \right) \\ &= \left( \prod_{i=0}^{h-1} x_i \right) \circ x_h \circ \left( \prod_{i=h+1}^{p-1} x_i \right) = \prod_{i=0}^{p-1} x_i, \end{align*}

which proves the theorem for this case. If $h = 0$ or $h = p - 1$ , the same result follows, but more simply, the terms arising from P or the terms arising from Q not appearing in the formulae.

Under a commutative associative law on a set E the composition of a finite family $(x_\alpha)_{\alpha \in A}$ of elements of E is by definition the common value of the composition of all the ordered sequences obtained by totally ordering A in all possible ways. This composition will still be denoted by $\prod_{\alpha \in A} x_\alpha$ under a law denoted by T; similarly for other notations.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Cites

Nothing.