Definition:Standard n Product

Definition

Let $\struct {S, \circ}$ be a semigroup.

Let $a_1, \ldots, a_n$ be a sequence of elements over $S$.

Then we denote the standard n product of $a_1, \ldots, a_n$ as:

$\ds \prod_{i \mathop = 1}^n a_i$

We define it inductively as follows:

If $n = 1$ then:

$\ds \prod_{i \mathop = 1}^1 a_i = a_1$

If $n > 1$ then:

$\ds \prod_{i \mathop = 1}^n a_i = \paren {\ds \prod_{i \mathop = 1}^{n - 1} a_i} a_n$

Sources

• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups