# Definition:Addition/Peano Structure

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## Contents

## Definition

Let $\struct {P, 0, s}$ be a Peano structure.

The binary operation $+$ is defined on $P$ as follows:

- $\forall m, n \in P: \begin{cases} m + 0 & = m \\ m + \map s n & = \map s {m + n} \end{cases}$

This operation is called **addition**.

The definition can equivalently be structured:

- $\forall m, n \in P: \begin{cases} 0 + n & = n \\ \map s m + n & = \map s {m + n} \end{cases}$

## Also defined as

Some treatments of Peano's axioms define the non-successor element (or **primal element**) to be $1$ and not $0$.

The treatments are similar, but the $1$-based system results in an algebraic structure which has no identity element for addition, and so no zero for multiplication.

Under this $1$-based system, **addition** is consequently defined as follows:

- $\forall m, n \in P: \begin{cases} m + 1 & = \map s m \\ m + \map s n & = \map s {m + n} \end{cases}$

or:

- $\forall m, n \in P: \begin{cases} 1 + n & = \map s n \\ \map s m + n & = \map s {m + n} \end{cases}$

## Also see

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 4$: The natural numbers - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 4$: Number systems $\text{I}$: Peano's Axioms