Definition 4

Let $(x_\alpha)_{\alpha \in A}$ be an ordered sequence of elements in a magma E whose indexing set A is non-empty. The composition (under the law T) of the ordered sequence $(x_\alpha)_{\alpha \in A}$, denoted by $\prod_{\alpha \in A} x_\alpha$, is the element of E defined by induction on the number of elements in A as follows:

(1) if $A = \{\beta\}$ then $\prod_{\alpha \in A} x_\alpha = x_\beta$;

(2) if A has $p > 1$ elements, $\beta$ is the least element of A and $A' = A - \{\beta\}$, then $\prod_{\alpha \in A} x_\alpha = x_\beta \ T \left( \prod_{\alpha \in A'} x_\alpha \right)$.

It follows immediately (by induction on the number of elements in the indexing sets) that the compositions of two similar ordered sequences are equal; in particular, the composition of any ordered sequence is equal to the composition of a finite sequence. If $A = \{\lambda, \mu, \nu\}$ has three elements ($\lambda < \mu < \nu$) the composition $\prod_{\alpha \in A} x_\alpha$ is $x_\lambda \ T \ (x_\mu \ T \ x_\nu)$.