Definition 3

Let $E$ be a unital magma. The identity element of $E$ is called the composition of the empty family of elements of $E$.

If $(x_\alpha)_{\alpha \in \varnothing}$ is the empty family of elements of E its composition $e$ is also denoted by $\prod_{\alpha \in \varnothing} x_\alpha$. For example, we write

when $p < q$ ($p, q \in \mathbf{N}$). Similarly we write $\prod^0 x = e$ for arbitrary $x$. With these definitions Theorems 1 and 3 of § 1 remain true if the hypothesis that the sets A and Bi are non-empty is suppressed.* Similarly the formulae $\prod^{m+n} x = \left( \prod^m x \right) \prod^n \left( \prod^n x \right)$ and $\prod^{mn} x = \prod^m \left( \prod^n x \right)$ are then true for $m \geq 0, n \geq 0$.

Let E be a unital magma whose law is denoted by $\top$ and $e$ its identity element. The support of a family $(x_i)_{i \in I}$ of elements of E is the set of indices $i \in I$ such that $x_i \neq e$. Let $(x_i)_{i \in I}$ be a family of elements of E with finite support. We shall define the composition $\prod_{i \in I} x_i$ in the two following cases:

(a) the set I is totally ordered; (b) E is associative and the $x_i$ are pairwise permutable.

In these two cases let S be the support of the family $(x_i)$. If J is a finite subset of I containing S, then $\prod_{i \in J} x_i = \prod_{i \in S} x_i$, as is seen by induction on the number of elements in J, applying Theorem 1 of § 1 in case (a) and Theorem 3 of § 1 in case (b). Let $\prod_{i \in I} x_i$ denote the common value of the compositions $\prod_{i \in J} x_i$ for all finite subsets of I containing S. When I is the interval $[p, \to]$ of $\mathbf{N}$, we also write $\prod_{i = p}^\infty x_i$.

With these definitions and notation, Theorems 1 and 3 of § 1 and the remarks following Theorem 3 extend to families with finite support.

The identity element of a law written additively is usually denoted by 0 and called zero or the null element (or sometimes the origin). Under a law written multiplicatively it is usually denoted by 1 and called the unit element (or unit).