Index Laws/Product of Indices/Notation
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Notation for Index Laws: Product of Indices
Let $\struct {S, \circ}$ be a semigroup.
Let $a \in S$.
Let $a^n$ be defined as the power of an element of a magma:
- $a^n = \begin{cases}
a : & n = 1 \\ a^x \circ a : & n = x + 1 \end{cases}$
that is:
- $a^n = \underbrace {a \circ a \circ \cdots \circ a}_{n \text{ instances of } a} = \circ^n \paren a$
Recall the index law for product of indices:
- $\circ^{n m} a = \circ^m \paren {\circ^n a} = \circ^n \paren {\circ^m a}$
This result can be expressed:
- $a^{n m} = \paren {a^n}^m = \paren {a^m}^n$
When additive notation $\struct {S, +}$ is used, the following is a common convention:
- $\paren {n m} a = m \paren {n a} = n \paren {m a}$
or:
- $\forall m, n \in \N_{>0}: \paren {n m} \cdot a = m \cdot \paren {n \cdot a} = n \cdot \paren {m \cdot a}$
Source of Name
The name index laws originates from the name index to describe the exponent $y$ in the power $x^y$.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Theorem $16.11$