Natural Numbers with Extension fulfil Naturally Ordered Semigroup Axioms 1, 3 and 4/Construction

Natural Numbers with Extension fulfil Naturally Ordered Semigroup Axioms 1, 3 and 4: Construction

Let $\N$ denote the set of natural numbers.

Let $\beta$ be an object such that $\beta \notin \N$

Let $M = \N \cup \set \beta$.

Let us extend the operation of natural number addition from $\N$ to $M$ by defining:

\(\ds 0 + \beta\)

\(=\)

\(\ds \beta + 0 = \beta\)

\(\ds \beta + \beta\)

\(=\)

\(\ds \beta\)

\(\ds n + \beta\)

\(=\)

\(\ds \beta + n = n\)

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.2$