Integers under Addition form Totally Ordered Group

Theorem

Let $\struct {\Z, +}$ denote the additive group of integers.

Let $\le$ be the usual ordering on $\Z$.

Then the ordered structure $\struct {\Z, +, \le}$ is a totally ordered group.

Proof

$\struct {\Z, +, \le}$ is an Ordered Structure

$(1)$

By Integer Addition is Closed, $\struct {\Z, +}$ is an algebraic structure.

$(2)$

$\le$ is an ordering on $\Z$.

Thus, $\struct {\Z, \le}$ is an ordered set.

$(3)$

By Ordering is Preserved on Integers by Addition and Integer Addition is Commutative, $\le$ is compatible with $+$.

Thus, $\struct {\Z, +, \le}$ is an ordered structure.

$\Box$

$\struct {\Z, +, \le}$ is a Totally Ordered Structure

By definition, the ordered structure $\struct {\Z, +, \le}$ is a totally ordered structure if and only if $\le$ is a total ordering.

This follows from Ordering on Integers is Total Ordering.

$\Box$

$\struct {\Z, +, \le}$ is a Totally Ordered Group

By definition, the totally ordered structure $\struct {\Z, +, \le}$ is a totally ordered group if and only if $\struct {\Z, +}$ is a group.

This follows from Integers under Addition form Abelian Group.

$\blacksquare$