Remark

Note that there is a certain arbitrariness about the definition of the composition of an ordered sequence; the induction we introduced proceeds "from right to left". If we proceeded "from left to right", the composition of the above ordered sequence $(x_\lambda, x_\mu, x_\nu)$ would be $(x_\lambda \ T \ x_\mu) \ T \ x_\nu$.

As a matter of notation, the composition of an ordered sequence $(x_\alpha)_{\alpha \in A}$ is written $\prod_{\alpha \in A}$ for a law denoted by $\perp$; for a law written additively it is usually denoted by $\sum_{\alpha \in A} x_\alpha$ and called the sum of the ordered sequence $(x_\alpha)_{\alpha \in A}$ (the $x_\alpha$ being called the terms of the sum); for a law written multiplicatively it is usually denoted by $\prod_{\alpha \in A} x_\alpha$ and called the product of the ordered sequence $(x_\alpha)$ (the $x_\alpha$ being called the factors of the product).†

When there is no possible confusion over the indexing set (nor over its ordering) it is often dispensed with in the notation for the composition of an ordered sequence and we then write, for example for a law written additively, $\sum_\alpha x_\alpha$ instead of $\sum_{\alpha \in A} x_\alpha$; similarly for the other notations.

For a law denoted by $\top$ the composition of a sequence $(x_i)$ with indexing set a non-empty interval $[p, q]$ of $\mathbf{N}$ is denoted by $\prod_{p \leq i \leq q} x_i$ or $\prod_{i=p}^q x_i$; similarly for laws denoted by other symbols.

Let E and F be two magmas whose laws of composition are denoted by $\top$ and $f$ a homomorphism of E into F. For every ordered sequence $(x_\alpha)_{\alpha \in A}$ of elements of E