Natural Numbers Bounded Below under Addition form Commutative Semigroup

Theorem

Let $m \in \N$ where $\N$ is the set of natural numbers.

Let $M \subseteq \N$ be defined as:

$M := \set {x \in \N: x \ge m}$

That is, $M$ is the set of all natural numbers greater than or equal to $m$.

Then the algebraic structure $\struct {M, +}$ is a commutative semigroup.

Proof

We have that:

Natural Number Addition is Associative

Natural Number Addition is Commutative

From Restriction of Associative Operation is Associative, $+$ is associative on $\struct {M, +}$.

From Restriction of Commutative Operation is Commutative, $+$ is commutative on $\struct {M, +}$.

It remains to be shown that $+$ is closed on $\struct {M, +}$.

Let $a, b \in M$.

Then $\exists r, s \in \N: a = m + r, b = m + s$.

Thus:

\(\ds a + b\)

\(=\)

\(\ds \paren {m + r} + \paren {m + s}\)

\(\ds \)

\(=\)

\(\ds m + \paren {m + r + s}\)

\(\ds \)

\(\ge\)

\(\ds m\)

So $a + b \in M$ and so $+$ is closed on $\struct {M, +}$.

$\blacksquare$