Corollary 1

Suppose that $L = \{1, m\}$; then the basis $(e_J)_{J \in \mathscr{P}(L)}$ of $\bigwedge(M)$ has $2^m$ elements. For $p > m$, $\bigwedge^p(M) = \{0\}$; $\bigwedge^m(M)$ has a basis consisting of a single element $e_L$; for $0 \leq p \leq m$ the number of elements in the basis $(e_J)$ of $\bigwedge^p(M)$ consisting of the $e_J$ such that $\mathrm{Card}(J) = p$ is

This follows from Set Theory, III, § 3, no. 5, Proposition 12 and Set Theory, III, § 5, no. 8, Corollary 1 to Proposition 11.

We return to the case where the set L in Theorem 1 is arbitrary and give explicitly the multiplication table (§ 1, no. 7) of the basis $(e_J)$. Given two finite subsets $J, K$ of the totally ordered set $L$, we write

where $v$ denotes the number of ordered pairs $(\lambda, \mu) \in J \times K$ such that $\lambda > \mu$. Then Corollary 1 to Proposition 4 of no. 2 immediately implies the relation Note the formula when $J \cap K = \varnothing, j = \mathrm{Card}(J), k = \mathrm{Card}(K)$ (no. 3, Corollary 2 to Proposition 5.)